Algebra Lineal Problemas Resueltos - María Isabel García Planas
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Transcript of Algebra Lineal Problemas Resueltos - María Isabel García Planas
8/10/2019 Algebra Lineal Problemas Resueltos - María Isabel García Planas
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A l g e b r a L i n e a l
P r o b l e m a s r e s u e l t o s
M
a
I s a b e l G a r c a P l a n a s
3
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Primera edición: septiembre de 1993Segunda edición: septiembre de 1994
Diseño de la cubieta: Antoni Gutiérrez
© M. Isabel GarcíaPlanas, 1993
© Edicions UPC, 1993
Edicions de la Universitat Politècnica de Catalunya, SLJordi Girona Salgado 31, 08034 BarcelonaTel.: 934 016 883 Fax: 934 015 885Edicions Virtuals: www.edicionsupc.ese-mail: [email protected]
Producción: Servei de Publicacions de la UPCy CPDAAV. Diagonal 647, ETSEIB. 08028 Barcelona
Depósito legal: B-22.363-93ISBN: 84-7653-295-4
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A ( J L )
2
S & M
a
I
5
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I N D I C E
C a p . 1 P o l i n o m i o s . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
C a p . 2 E s p a c i o s v e c t o r i a l e s . . . . . . . . . . . . . . . . . . . . . . 2 3
C a p . 3 S i s t e m a s d e e c u a c i o n e s . M a t r i c e s . . . . . . . . . . . . . . . 3 9
C a p . 4 A p l i c a c i o n e s l i n e a l e s . . . . . . . . . . . . . . . . . . . . . 5 1
C a p . 5 D e t e r m i n a n t e s . . . . . . . . . . . . . . . . . . . . . . . . 7 3
C a p . 6 D i a g o n a l i z a c i o n d e e n d o m o r s m o s . . . . . . . . . . . . . . 8 5
C a p . 7 F o r m a r e d u c i d a d e J o r d a n . . . . . . . . . . . . . . . . . . 9 9
C a p . 8 A n a l i s i s m a t r i c i a l . . . . . . . . . . . . . . . . . . . . . . . 1 1 7
A p e n d i c e I G r u p o s . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 1
A p e n d i c e I I A n i l l o d e c l a s e s d e r e s t o . . . . . . . . . . . . . . . . . 1 4 1
9
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P o l i n o m i o s y F r a c c i o n e s r a c i o n a l e s 1 1
C a p t u l o 1 P o l i n o m i o s y f r a c c i o n e s r a c i o n a l e s
1 H a l l a r e l m a x i m o c o m u n d i v i s o r , p o r e l a l g o r i t m o d e E u c l i d e s , d e l o s p o l i n o m i o s
P
1
( x ) = 2 1 5 6 x
5
+ 1 1 2 0 x
4
; 4 3 3 x
3
; 1 7 9 x
2
+ 3 2 x + 4
P
2
( x ) = 1 3 7 2 x
5
+ 7 8 4 x
4
; 2 4 5 x
3
; 1 3 1 x
2
+ 1 6 x + 4
S o l u c i o n :
R e c o r d a n d o e l t e o r e m a d e E u c l i d e s :
M C D ( P
1
( x ) P
2
( x ) ) = M C D ( P
2
( x ) R ( x ) )
S i e n d o R ( x ) e l r e s t o d e d i v i d i r P
1
( x ) e n t r e P
2
( x )
S a b e m o s q u e
M C D ( P
1
( x ) P
2
( x ) ) = M C D ( P
1
( x ) P
2
( x ) ) 8 u n i d a d e n R x
y a l s e r 2 1 5 6 = 4 7
2
1 1 y 1 3 7 2 = 4 7
3
, m u l t i p l i c a r e m o s P
1
( x ) p o r 7 p a r a e v i t a r
f r a c c i o n e s a l h a c e r l a d i v i s i o n d e P
1
( x ) p o r P
2
( x )
7 P
1
( x ) = P
2
( x ) 1 1 + ( ; 7 8 4 x
4
; 3 3 6 x
3
+ 1 8 8 x
2
+ 4 8 x ; 1 6 )
R ( x ) = ; 7 8 4 x
4
; 3 3 6 x
3
+ 1 8 8 x
2
+ 4 8 x ; 1 6
q u e s i m p l i c a m o s p o r ; 4 q u e d a n d o
R ( x ) = 1 9 6 x
4
+ 8 4 x
3
; 4 7 x
2
; 1 2 x + 4
P
2
( x ) = R ( x ) ( 7 x + 1 ) + 0 l u e g o M C D ( P
2
( x ) R ( x ) ) = R ( x )
p o r l o q u e :
M C D ( P
1
( x ) P
2
( x ) ) = R ( x ) = 1 9 6 x
4
+ 8 4 x
3
; 4 7 x
2
; 1 2 x + 4
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1 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
2 H a l l a r l a s r a c e s d e l p o l i n o m i o P ( x ) = x
4
; x
3
; 3 x
2
+ 5 x ; 2 s a b i e n d o q u e
u n a d e e l l a s e s t r i p l e .
S o l u c i o n :
L a d e s c o m p o s i c i o n e n f a c t o r e s p r i m o s d e l p o l i n o m i o s e r a :
P ( x ) = ( x ; )
3
( x ; )
S i e s u n a r a z t r i p l e d e P ( x ) e s r a z d o b l e d e P
0
( x ) y s i m p l e d e P " ( x )
P o r l o t a n t o e l M C D ( P
0
( x ) P " ( x ) ) c o n t i e n e e l f a c t o r ( x ; ) . B a s t a p u e s h a l l a r
M C D ( P
0
( x ) P " ( x ) ) y e n t r e s u s f a c t o r e s , p o r t a n t e o e n P ( x ) , p u e d e e x t r a e r s e e l v a l o r
d e
A h o r a b i e n , e n e s t e c a s o c o n c r e t o , p u e s t o q u e P " ( x ) e s d e g r a d o d o s , r e s u l t a m a s
s e n c i l l o h a l l a r l a s r a c e s d e P " y d e l a s d o s v e r c u a l l o e s t a m b i e n d e P ( x )
P
0
( x ) = 4 x
3
; 3 x
2
; 6 x + 5
P " ( x ) = 1 2 x
2
; 6 x ; 6
D e P " ( x ) = 0 t e n e m o s x = ;
1
2
x = 1
P ( ;
1
2
) 6= 0 , l u e g o ;
1
2
n o e s r a z d e P ( x ) , s i n e m b a r g o P ( 1 ) = 0 l u e g o = 1 e s l a
r a z b u s c a d a .
P u e s t o q u e d a d o u n p o l i n o m i o p ( x ) = a
n
x
n
+ + a
0
c o n r a c e s
1
: : :
n
c o n t a d a s
c o n s u m u l t i p l i c i d a d , e s a
n ; 1
= ; (
1
+ +
n
) , s e t i e n e = ; 2
3 P r o b a r q u e P ( x ) = n x
n + 2
; ( n + 2 ) x
n + 1
+ ( n + 2 ) x ; n e s d i v i s i b l e p o r ( x ; 1 )
3
( S e s u p o n e P ( x ) 2 R x y n 2 N )
S o l u c i o n :
Q u e P ( x ) s e a d i v i s i b l e p o r ( x ; 1 )
3
e q u i v a l e a q u e 1 e s p o r l o m e n o s r a z t r i p l e
d e P ( x ) , r a z d o b l e p o r l o m e n o s , d e P
0
( x ) y r a z s i m p l e p o r l o m e n o s , d e P " ( x )
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P o l i n o m i o s y F r a c c i o n e s r a c i o n a l e s 1 3
V e a m o s
P ( 1 ) = n ; ( n + 2 ) + ( n + 2 ) ; n = 0 l u e g o 1 e s r a z d e P ( x )
P
0
( x ) = n ( n + 2 ) x
n + 1
; ( n + 2 ) ( n + 1 ) x
n
+ ( n + 2 )
P
0
( 1 ) = n ( n + 2 ) ; ( n + 2 ) ( n + 1 ) + ( n + 2 ) = 0 l u e g o 1 e s r a z d e P
0
( x )
P " ( x ) = n ( n + 2 ) ( n + 1 ) x
n
; ( n + 2 ) ( n + 1 ) n x
n ; 1
P " ( 1 ) = n ( n + 2 ) ( n + 1 ) ; ( n + 2 ) ( n + 1 ) n = 0 l u e g o 1 e s r a z d e P " ( x )
p o r l o t a n t o P ( x ) e s d i v i s i b l e p o r ( x ; 1 )
3
c o m o p r e t e n d a m o s p r o b a r
O b s e r v a m o s a d e m a s q u e P ( x ) n o e s d i v i s i b l e p o r ( x ; 1 )
4
p u e s
P
0 0 0
( x ) = n
2
( n + 2 ) ( n + 1 ) x
n ; 1
; ( n + 2 ) ( n + 1 ) ( n ; 1 ) n x
n ; 2
P
0 0 0
( 1 ) = n
2
( n + 2 ) ( n + 1 ) ; ( n + 2 ) ( n + 1 ) ( n ; 1 ) n = n ( n + 1 ) ( n + 2 ) 6= 0
4 C o n s i d e r e m o s P ( x ) = x
3
; 4 x
2
+ 5 x ; 2 a c o e c i e n t e s r e a l e s .
a ) D e t e r m i n a r P
0
( x ) ( p o l i n o m i o d e r i v a d o d e P ( x ) ) y d a r s u d e s c o m p o s i c i o n e n
f a c t o r e s p r i m o s .
b ) P r o b a r q u e u n a d e l a s r a c e s d e P
0
( x ) l o e s t a m b i e n d e P ( x ) y d e d u c i r d e e s t o
l a d e s c o m p o s i c i o n e n f a c t o r e s p r i m o s d e P ( x )
c ) C a l c u l a r M C D ( P ( x ) P
0
( x ) ) y d e t e r m i n a r d o s p o l i n o m i o s P
1
( x ) y P
2
( x ) t a l e s
q u e :
P
1
( x ) P ( x ) + P
2
( x ) P
0
( x ) = M C D ( P ( x ) P
0
( x ) )
S o l u c i o n :
a )
P
0
( x ) = 3 x
2
; 8 x + 5 = ( x ; 1 ) ( 3 x ; 5 )
b ) L a s r a c e s d e P
0
( x ) s o n 1 y
5
3
, a h o r a b i e n :
P (
5
3
) 6= 0 l u e g o
5
3
n o e s r a z d e P ( x )
P ( 1 ) = 0 l u e g o 1 e s r a z d o b l e d e P ( x )
P " ( 1 ) = ; 2 l u e g o 1 n o e s r a z t r i p l e d e P ( x )
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1 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
L u e g o
P ( x ) = ( x ; 1 )
2
( x ; a ) = x
3
; ( 2 + a ) x
2
+ ( 2 a + 1 ) x ; a =
= x
3
; 4 x
2
+ 5 x ; 2
d e d o n d e a = 2
c ) d e
P ( x ) = ( x ; 1 )
2
( x ; 2 )
P
0
( x ) = ( x ; 1 ) ( 3 x ; 5 )
s e d e d u c e q u e : M C D ( P ( x ) P
0
( x ) ) = ( x ; 1 )
P o r e l a l g o r i t m o d e d i v i s i o n ( P ( x ) = P
0
( x ) Q ( x ) + R ( x ) ) t e n e m o s :
9 P ( x ) = P
0
( x ) ( 3 x ; 4 ) + ( ; 2 x + 2 ) = P
0
( x ) ( 3 x ; 4 ) ; 2 ( x ; 1 )
D e s p e j a n d o ( x ; 1 )
9 P ( x ) ; P
0
( x ) ( 3 x ; 4 ) = ; 2 ( x ; 1 )
; 9
2
P ( x ) ;
1
2
( 3 x ; 4 ) P
0
( x ) = ( x ; 1 )
L u e g o
P
1
( x ) =
; 9
2
P
2
( x ) =
; 1
2
( 3 x ; 4 )
5 L o s r e s t o s d e d i v i d i r u n p o l i n o m i o P ( x ) 2 R x ] p o r x ; 1 x ; 2 y x ; 3
s o n r e s p e c t i v a m e n t e 3 7 1 3
D e t e r m i n a r e l r e s t o d e d i v i d i r P ( x ) p o r e l p r o d u c t o
( x ; 1 ) ( x ; 2 ) ( x ; 3 )
S o l u c i o n :
P o r e l a l g o r i t m o d e d i v i s i o n s a b e m o s
P ( x ) = D ( x ) Q ( x ) + R ( x ) c o n g r a d o R ( x ) < g r a d o D ( x )
P ( x ) = ( x ; 1 ) Q
1
( x ) + R
1
( x ) = ( x ; 1 ) Q
1
( x ) + 3
P ( x ) = ( x ; 2 ) Q
2
( x ) + R
2
( x ) = ( x ; 2 ) Q
2
( x ) + 7
P ( x ) = ( x ; 3 ) Q
3
( x ) + R
3
( x ) = ( x ; 3 ) Q
3
( x ) + 1 3
P ( x ) = ( x ; 1 ) ( x ; 2 ) ( x ; 3 ) Q + R ( x ) c o n R ( x ) = a x
2
+ b x + c
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P o l i n o m i o s y F r a c c i o n e s r a c i o n a l e s 1 5
S a b e m o s q u e e l v a l o r n u m e r i c o d e u n p o l i n o m i o P ( x ) e n u e s e l r e s t o d e
d i v i d i r e l p o l i n o m i o p o r x ; u l u e g o y p a r a i = 1 2 3
P ( u ) = ( x ; u ) Q
i
( u ) + R
i
( u ) = ( u ; 1 ) ( u ; 2 ) ( u ; 3 ) Q ( u ) + R ( u )
d e d o n d e :
P ( 1 ) = R
1
( 1 ) = 3 = R ( 1 ) = a + b + c
P ( 2 ) = R
2
( 2 ) = 7 = R ( 2 ) = 4 a + 2 b + c
P ( 3 ) = R
3
( 3 ) = 1 3 = R ( 3 ) = 9 a + 3 b + c
9
>
=
>
q u e , r e s o l v i e n d o e l s i s t e m a n o s q u e d a :
a = b = c = 1 y R ( x ) = x
2
+ x + 1
6 E n c o n t r a r u n p o l i n o m i o P ( x ) 2 R x ] d e g r a d o c i n c o , t a l q u e P ( x ) + 1 0 s e a
d i v i s i b l e p o r ( x + 2 )
3
y P ( x ) ; 1 0 s e a d i v i s i b l e p o r ( x ; 2 )
3
S o l u c i o n :
P u e s t o q u e P ( x ) + 1 0 e s d i v i s i b l e p o r ( x + 2 )
3
, t e n e m o s q u e P
0
( x ) =
( P ( x ) + 1 0 )
0
e s d i v i s i b l e p o r ( x + 2 )
2
y p u e s t o q u e P ( x ) ; 1 0 e s d i v i s i b l e
p o r ( x ; 2 )
3
, t e n e m o s q u e P
0
( x ) = ( P ( x ) ; 1 0 )
0
e s d i v i s i b l e p o r ( x ; 2 )
2
l u e g o P
0
( x ) ( p o l i n o m i o d e g r a d o c u a t r o ) s e r a
P
0
( x ) = k ( x + 2 )
2
( x ; 2 )
2
= k ( x
4
; 8 x
2
+ 1 6 ) c o n k 2 R
d e d o n d e
P ( x ) = k (
x
5
5
;
8
3
x
3
+ 1 6 x + c ) c o n c 2 R e i m p o n i e n d o q u e
P ( ; 2 ) = ; 1 0
P ( 2 ) = 1 0
N o t a : O b s e r v a m o s q u e
P ( x ) + 1 0 = ( x + 2 )
3
Q
1
( x ) = ) P ( x ) = ( x + 2 )
3
Q
1
( x ) ; 1 0 = ) P ( ; 2 ) = ; 1 0
P ( x ) ; 1 0 = ( x ; 2 )
3
Q
2
( x ) = ) P ( x ) = ( x ; 2 )
3
Q
2
( x ) + 1 0 = ) P ( 2 ) = 1 0
Primera edición: septiembre de 1993
8/10/2019 Algebra Lineal Problemas Resueltos - María Isabel García Planas
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1 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
r e s u l t a
; 1 0 = k (
; 3 2
5
+
6 4
3
; 3 2 + c )
1 0 = k (
3 2
5
;
6 4
3
+ 3 2 + c )
y r e s o l v i e n d o e l s i s t e m a t e n e m o s
c = 0 k =
7 5
1 2 8
y P ( x ) =
1 5
1 2 8
x
5
;
2 5
1 6
x
3
+
7 5
8
x
O t r o m e t o d o :
D e :
P ( x ) + 1 0 = ( x + 2 )
3
C
1
( x ) c o n g r a d o C
1
( x ) = 2
P ( x ) ; 1 0 = ( x + 2 )
3
C
2
( x ) c o n g r a d o C
2
( x ) = 2
t e n e m o s :
2 0 = ( x + 2 )
3
C
1
( x ) ; ( x ; 2 )
3
C
2
( x )
1 = ( x + 2 )
3
(
1
2 0
C
1
) + ( x ; 2 )
3
(
1
2 0
C
2
( x ) )
e s d e c i r , h e m o s d e b u s c a r
1
2 0
C
1
( x ) y
1
2 0
C
2
( x ) q u e s o n l o s p o l i n o m i o s d e g r a d o m n i m o
q u e h a c e n q u e s e c u m p l a l a i d e n t i d a d d e B e z o u t , ( o b s e r v e s e q u e ( x + 2 )
3
y ( x ; 2 )
3
s o n p r i m o s e n t r e s ) .
7 D e s c o m p o n e r e n f r a c c i o n e s s i m p l e s s o b r e R , l a f r a c c i o n
; 1 4 x
2
+ 3 x ; 3 9
( x ; 1 )
2
( x ; 3 ) ( x
2
+ 4 )
I d e m s o b r e C
S o l u c i o n :
P l a n t e a m o s
; 1 4 x
2
+ 3 x ; 3 9
( x ; 1 )
2
( x ; 3 ) ( x
2
+ 4 )
=
a
x ; 1
+
b
( x ; 1 )
2
+
c
x ; 3
+
d x + e
x
2
+ 4
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P o l i n o m i o s y F r a c c i o n e s r a c i o n a l e s 1 7
( O b s e r v a m o s q u e x
2
+ 4 e s p r i m o s o b r e R ) . O p e r a n d o e n e l s e g u n d o m i e m b r o , q u e d a
; 1 4 x
2
+ 3 x ; 3 9
( x ; 1 )
2
( x ; 3 ) ( x
2
+ 4 )
=
a ( x ; 1 ) ( x ; 3 ) ( x
2
+ 4 ) + b ( x ; 3 ) ( x
2
+ 4 ) + c ( x ; 1 )
2
( x
2
+ 4 ) + ( d x + e ) ( x ; 1 )
2
( x ; 3 )
( x ; 1 )
2
( x ; 3 ) ( x
2
+ 4 )
D e l a i g u a l d a d d e e s t a s d o s f r a c c i o n e s s e d e d u c e l a i g u a l d a d d e l o s p o l i n o m i o s n u -
m e r a d o r e s d e l a s f r a c c i o n e s . D e a q u s e d e d u c e p o r l o t a n t o u n m e t o d o d e c a l c u l o
d e l o s c o e c i e n t e s d e s c o n o c i d o s : i d e n t i c a r l o s c o e c i e n t e s d e i g u a l g r a d o d e a m b o s
p o l i n o m i o s , o b t e n i e n d o a s u n s i s t e m a d e c i n c o e c u a c i o n e s c o n c i n c o i n c o g n i t a s . O t r o
m e t o d o m a s s e n c i l l o p a r a o b t e n e r l o s c o e c i e n t e s e s : s i d o s p o l i n o m i o s s o n i g u a l e s , s u s
f u n c i o n e s p o l i n o m i c a s a s o c i a d a s s o n t a m b i e n i g u a l e s , p o r l o q u e , d a n d o v a l o r e s a x e n
a m b o s p o l i n o m i o s , l o s v a l o r e s n u m e r i c o s h a n d e s e r i g u a l e s . A s ,
p a r a x = 1 e s ; 1 4 + 3 ; 3 9 = b ( 1 ; 3 ) ( 1 + 4 ) ) b = 5
p a r a x = 3 e s ; 1 4 3 2 + 9 ; 3 9 = c ( 3 ; 1 )
2
( 3
2
+ 4 ) ) c = ; 3
p a r a x = 0 e s ; 3 9 = 1 2 a ; 6 0 ; 1 2 ; 3 e ) 1 2 a ; 3 e = 3 3
p a r a x = ; 1 e s ; 1 4 ; 3 ; 3 9 = 4 0 a ; 1 0 0 ; 6 0 ; 1 6 ( ; d + e ) ) 1 0 a + 4 d ; 4 e = 2 6
p a r a x = 2 e s ; 5 6 + 6 ; 3 9 = ; 8 a ; 4 0 ; 2 4 ; ( 2 d + e ) ) 8 a + 2 d + e = 2 5
R e s o l v i e n d o l a s t r e s u l t i m a s e c u a c i o n e s r e s u l t a a = 3 , d = 0 , e = 1 , l u e g o l a
d e s c o m p o s i c i o n e s
3
x ; 1
+
5
( x ; 1 )
2
+
; 3
x ; 3
+
1
x
2
+ 4
P a s e m o s a h o r a a e f e c t u a r l a d e s c o m p o s i c i o n e n C
x
2
+ 4 y a n o e s p r i m o e n C , x
2
+ 4 = ( x ; 2 i ) ( x + 2 i ) , p o r l o q u e l a d e s c o m p o s i c i o n
s e r a :
; 1 4 x
2
+ 3 x ; 3 9
( x ; 1 )
2
( x ; 3 ) ( x
2
+ 4 )
=
a
x ; 1
+
b
( x ; 1 )
2
+
c
x ; 3
+
m
x ; 2 i
+
n
x + 2 i
C o m p a r a n d o e s t a d e s c o m p o s i c i o n c o n l a a n t e r i o r , p o d e m o s a s e g u r a r q u e a , b , c
s e r a n l o s m i s m o s o b t e n i d o s p a r a e l c a s o r e a l , y
m
x ; 2 i
+
n
x + 2 i
=
1
x
2
+ 4
p o r l o q u e
1 = m ( x + 2 i ) + n ( x ; 2 i ) , q u e p a r a x = ; 2 i s e t i e n e 1 = ; 4 n i ! n = +
1
4
i
p a r a x = 2 i s e t i e n e 1 = 4 m i ) m = ;
1
4
i , y l a d e s c o m p o s i c i o n e s
3
x ; 1
+
5
( x ; 1 )
2
+
; 3
x ; 3
+
;
1
4
i
x ; 2 i
+
1
4
i
x + 2 i
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1 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
8 D e s c o m p o n e r e n f r a c c i o n e s s i m p l e s s o b r e C , R y Q l a f r a c c i o n r a c i o n a l s i g u i e n t e
t
6
+ t
4
; t
2
; 1
t
3
( t
2
; 2 t ; 1 )
= Q ( t )
S o l u c i o n :
P u e s t o q u e e l g r a d o d e l n u m e r a d o r e s m a y o r q u e e l d e l d e n o m i n a d o r , e f e c t u a m o s l a
d i v i s i o n y t e n e m o s
Q ( t ) = t + 2 +
6 t
4
+ 2 t
3
; t
2
; 1
t
5
; 2 t
4
; t
3
t
5
; 2 t
4
; t
3
t i e n e l a m i s m a d e s c o m p o s i c i o n e n f a c t o r e s p r i m o s t a n t o s o b r e R c o m o
s o b r e C :
t
5
; 2 t
4
; t
3
= t
3
( t ; 1 ;
p
2 ) ( t ; 1 +
p
2 )
N o a s s o b r e Q q u e t
2
; 2 t ; 1 e s p r i m o , p o r l o q u e l a d e s c o m p o s i c i o n e n f r a c c i o n e s
s i m p l e s d e Q ( t ) s e r a l a m i s m a t a n t o s o b r e R c o m o s o b r e C y d i s t i n t a p a r a Q
V e a m o s p a r a R y C :
Q ( t ) = t + 2 +
A
t
3
+
B
t
2
+
C
t
+
D
t ; 1 ;
p
2
+
E
t ; 1 +
p
2
q u e o p e r a n d o o b t e n e m o s
Q ( t ) = t + 2 +
( A + B t + C t
2
) ( t
2
; 2 t ; 1 ) + t
3
( D ( t ; 1 +
p
2 ) + E ( t ; 1 ;
p
2 ) )
t
5
; 2 t
4
; t
3
= t + 2 +
P ( t )
t
5
; 2 t
4
; t
3
p o r l o q u e
6 t
4
+ 2 t
3
; t
2
; 1 = ( A + B t + C t
2
) ( t
2
; 2 t ; 1 ) + t
3
( D ( t ; 1 +
p
2 ) + E ( t ; 1 ;
p
2 ) ) = P ( t )
y h a c i e n d o u s o d e l h e c h o : s i d o s p o l i n o m i o s s o n i g u a l e s t a m b i e n l o s o n s u s f u n c i o n e s
p o l i n o m i c a s a s o c i a d a s , t e n e m o s
( 6 t
4
+ 2 t 3 ; t
2
; 1 ) ( 0 ) = ; 1 = P ( 0 ) = ; A
( 6 t
4
+ 2 t
3
; t
2
; 1 )
0
( 0 ) = 0 = P
0
( 0 ) = ; B ; 2 A
( 6 t
4
+ 2 t 3 ; t
2
; 1 )
0 0
( 0 ) = ; 2 ; P
0 0
( 0 ) = 2 ( A ; 2 B ; C )
( 6 t
4
+ 2 t
3
; t
2
; 1 )
0 0 0
( 0 ) = 1 2 =
= P
0 0 0
( 0 ) = 6 ( B ; 2 C + ( ; 1 +
p
2 ) D + ( ; 1 ;
p
2 ) E )
( 6 t
4
+ 2 t
3
; t
2
; 1 )
0 0 0 0
( 0 ) = 1 4 4 = P
0 0 0 0
( 0 ) = 4 8 C + 4 8 ( D + E )
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P o l i n o m i o s y F r a c c i o n e s r a c i o n a l e s 1 9
R e s o l v i e n d o e l s i s t e m a d e e c u a c i o n e s o b t e n e m o s :
A = 1 B = ; 2 C = 6 D = 4
p
2 E = ; 4
p
2
y l a d e s c o m p o s i c i o n e s
Q ( t ) = t + 2 +
1
t
3
+
; 2
t
2
+
6
t
+
4
p
2
t ; 1 ;
p
2
+
; 4
p
2
t ; 1 +
p
2
P a s e m o s a l a d e s c o m p o s i c i o n d e Q ( t ) s o b r e Q :
Q ( t ) = t + 2 +
A
t
3
+
B
t
2
+
C
t
+
M
t
2
; 2 t ; 1
h a c i e n d o
4
p
2
t ; 1 ;
p
2
+
; 4
p
2
t ; 1 +
p
2
=
1 6
t
2
; 2 t ; 1
2 Q ( t )
p o r l o q u e
Q ( t ) = t + 2 +
1
t
3
+
; 2
t
2
+
6
t
+
1 6
t
2
; 2 t ; 1
y p u e s t o q u e l a d e s c o m p o s i c i o n e n f r a c c i o n e s s i m p l e s e s u n i c a , e s t a s e r a l a d e s c o m -
p o s i c i o n s o b r e Q
9 D e s c o m p o n e r e n f r a c c i o n e s s i m p l e s s o b r e C l a f r a c c i o n r a c i o n a l s i g u i e n t e
Q ( x ) =
1
( x ; 3 )
9
( x ; 5 )
9
S o l u c i o n :
L a d e s c o m p o s i c i o n s e r a
Q ( x ) =
9
X
n = 1
A
n
( x ; 3 )
n
+
9
X
n = 1
B
n
( x ; 5 )
n
d o n d e A
n
, B
n
c o n n = 1 : : : 9 s o n n u m e r o s c o m p l e j o s a d e t e r m i n a r .
C o n s i d e r e m o s F ( x ) =
1
( x ; 5 )
9
f u n c i o n r a c i o n a l d e s a r r o l l a m o s F ( x ) p o r l a f o r m u l a
d e T a y l o r e n e l p u n t o x = 3 , h a s t a e l o r d e n 8 , o b t e n i e n d o
F ( x ) = F ( 3 ) +
F
0
( 3 )
1 !
( x ; 3 ) + +
F
8
( 3 )
8 !
( x ; 3 )
8
+ G ( x ) ( x ; 3 )
9
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2 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
s i e n d o G ( x ) u n a f u n c i o n r a c i o n a l q u e e s t a d e n i d a p a r a x = 3 u s a n d o e s t e d e s a r r o l l o
t e n e m o s
1
( x ; 3 )
9
( x ; 5 )
9
=
F ( 3 )
( x ; 3 )
9
+
F
0
( 3 )
( x ; 3 )
8
+ +
F
8
( 3 )
8 ! ( x ; 3 )
+ G ( x )
P o r l a u n i c i d a d d e l o s c o e c i e n t e s A
n
y B
n
t e n e m o s
A
n
=
F
9 ; n
( 3 )
( 9 ; n ) !
F
n
( x ) = ( ; 9 ) ( ; 9 ; 1 ) ( ; 9 ; n + 1 ) ( x ; 5 )
; 9 ; n
= ( ; 1 )
n
( 8 + n ) !
8 !
1
( x ; 5 )
9 + n
l u e g o F
n
( 3 ) = ( ; 1 )
n
( 8 + n ) !
8 !
1
( ; 2 )
9 + n
y A
n
= ( ; 1 )
9 ; n
( 1 7 ; n ) !
8 ! ( 9 ; n ) !
1
( ; 2 )
1 8 ; n
y p o r s i m e t r a t e n e m o s , ( o b s e r v e s e q u e o b t e n e m o s B
n
c o n s i d e r a n d o F
1
( x ) =
1
( x ; 3 )
n
y r e p i t i e n d o e l p r o c e s o a n t e r i o r ) .
B
n
= ( ; 1 )
9 ; n
( 1 7 ; n ) !
8 ! ( 9 ; n ) !
1
2
1 8 ; n
1 0 S o b r e e l c u e r p o d e l o s r a c i o n a l e s , d e s c o m p o n e r e n f r a c c i o n e s s i m p l e s l a f r a c c i o n
r a c i o n a l s i g u i e n t e
Q ( x ) =
2 ( x
2
+ 1 )
( x + 1 ) ( x
3
+ 2 )
S o l u c i o n :
O b s e r v a m o s q u e x
3
+ 2 n o t i e n e r a c e s e n Q , l u e g o
2 ( x
2
+ 1 )
( x + 1 ) ( x
3
+ 2 )
=
A
( x + 1 )
+
B x
2
+ C x + D
( x
3
+ 2 )
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P o l i n o m i o s y F r a c c i o n e s r a c i o n a l e s 2 1
O p e r a n d o
2 ( x
2
+ 1 )
( x + 1 ) ( x
3
+ 2 )
=
A ( x
3
+ 2 ) + ( B x
2
+ C x + D ) ( x + 1 )
( x + 1 ) ( x
3
+ 2 )
I g u a l a n d o n u m e r a d o r e s t e n e m o s
A + B = 0
B + C = 2
C + D = 0
2 A + D = 2
9
>
>
>
=
>
>
>
)
A = 4
B = ; 4
C = 6
D = ; 6
l u e g o l a d e s c o m p o s i c i o n e s
Q ( x ) =
4
( x + 1 )
+
; 4 x
2
+ 6 x ; 6
x
3
+ 2
1 1 D e s c o m p o n e r s o b r e R l a f r a c c i o n :
Q ( x ) =
x
2
( x
2
+ 1 )
n
S o l u c i o n :
H a c i e n d o x
2
+ 1 = y t e n e m o s x
2
= y ; 1 , l u e g o
x
2 n
( x
2
+ 1 )
n
=
( y ; 1 )
n
y
n
=
P
n
i = 0
n
i
( ; 1 )
i
y
n ; i
y
n
=
= 1 ;
n
1
y
+
n
2
y
2
+ + ( ; 1 )
n
n
1
y
=
= 1 ;
n
1
x
2
+ 1
+
n
2
( x
2
+ 1 )
2
+ + ( ; 1 )
n
n
n
( x
2
+ 1 )
n
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E s p a c i o s v e c t o r i a l e s 2 3
C a p t u l o 2 E s p a c i o s v e c t o r i a l e s
1 S e a R
0
e l g r u p o m u l t i p l i c a t i v o d e l o s n u m e r o s r e a l e s e s t r i c t a m e n t e p o s i t i v o s . P r o -
b a r q u e R
0
R
0
R
0
e s u n R - e s p a c i o v e c t o r i a l c o n l a s o p e r a c i o n e s s i g u i e n t e s :
8 ( x y z ) ( x
1
y
1
z
1
) 2 R
0
R
0
R
0
8 2 R
( x y z ) ( x
1
y
1
z
1
) = ( x x
1
y y
1
z z
1
)
( x y z ) = ( x
y
z
)
E n c a s o d e s e r d i m e n s i o n n i t a d e t e r m i n a r u n a b a s e
S o l u c i o n :
E s f a c i l p r o b a r q u e c o n l a o p e r a c i o n e l c o n j u n t o R
0
R
0
R
0
e s u n g r u p o a b e l i a n o :
A s o c i a t i v i d a d
8 ( x y z ) ( x
1
y
1
z
1
) ( x
2
y
2
z
2
) 2 R
0
R
0
R
0
( x y z ) ( ( x
1
y
1
z
1
) ( x
2
y
2
z
2
) ) = ( x y z ) ( x
1
x
2
y
1
y
2
z
1
z
2
) =
= ( x ( x
1
x
2
) y y
1
y
2
) z z
1
z
2
) ) = ( ( x x
1
) x
2
( y y
1
) y
2
( z z
1
) z
2
) =
= ( x x
1
y y
1
z z
1
) ( x
2
y
2
z
2
) = ( ( x y z ) ( x
1
y
1
z
1
) ) ( x
2
y
2
z
2
)
( E s t a p r o p i e d a d n o s p e r m i t e e s c r i b i r ( x y z ) ( x
1
y
1
z
1
) ( x
2
y
2
z
2
) )
C o n m u t a t i v i d a d
8 ( x y z ) ( x
1
y
1
z
1
) 2 R
0
R
0
R
0
( x y z ) ( x
1
y
1
z
1
) = ( x x
1
y y
1
z z
1
) = ( x
1
x y
1
y z
1
z ) =
= ( x
1
y
1
z
1
) ( x y z )
E l e m e n t o n e u t r o
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2 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
8 ( x y z ) 2 R
0
R
0
R
0
( 1 1 1 ) ( x y z ) = ( 1 x 1 y 1 z ) = ( x y z )
E l e m e n t o s i m e t r i c o
8 ( x y z ) 2 R
0
R
0
R
0
9 ( 1 = x 1 = y 1 = z ) 2 R
0
R
0
R
0
t a l q u e
( x y z ) ( 1 = x 1 = y 1 = z ) = ( x 1 = x y 1 = y : z 1 = z ) = ( 1 1 1 )
V e a m o s a h o r a q u e l a o p e r a c i o n e x t e r n a v e r i c a l a s c u a t r o p r o p i e d a d e s n e c e s a r i a s p a r a
q u e e l c o n j u n t o s e a u n e s p a c i o v e c t o r i a l :
P r i m e r a l e y d i s t r i b u t i v a
8 2 R 8 ( x y z ) ( x
1
y
1
z
1
) 2 R
0
R
0
R
0
( ( x y z ) ( x
1
y
1
z
1
) ) = ( x x
1
y y
1
z z
1
) =
= ( ( x x
1
)
( y y
1
)
( z z
1
)
) = ( x
x
1
y
y
1
z
z
1
) =
= ( x
y
z
) ( x
1
y
1
z
1
) = ( ( x y z ) ) ( ( x
1
y
1
z
1
) )
S e g u n d a l e y d i s t r i b u t i v a
8 2 R 8 ( x y z ) 2 R
0
R
0
R
0
( + ) ( x y z ) = ( x
+
y
+
z
+
) = ( x
x
y
y
z
z
) =
= ( x
y
z
) ( x
y
z
) = ( ( x y z ) ) ( ( x y z ) )
A s o c i a t i v i d a d d e l o s e s c a l a r e s
8 2 R 8 ( x y z ) 2 R
0
R
0
R
0
( ) ( x y z ) = ( x
y
z
) =
= ( ( x
)
( y
)
( z
)
) = ( x
y
z
) =
= ( ( x y z ) )
P r o p i e d a d d e l e l e m e n t o u n i d a d d e l c u e r p o
8 ( x y z ) 2 R
0
R
0
R
0
1 ( x y z ) = ( x
1
y
1
z
1
) = ( x y z )
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E s p a c i o s v e c t o r i a l e s 2 5
l u e g o , e n e f e c t o R
0
R
0
R
0
e s u n R - e s p a c i o v e c t o r i a l .
V e a m o s c u a l e s s u d i m e n s i o n y s i e s p o s i b l e , d e t e r m i n e m o s u n a b a s e .
S a b e m o s q u e 8 x 2 R
0
x = e
o g x
, l u e g o 8 ( x y z ) 2 R
0
R
0
R
0
, s e t i e n e
( x y z ) = ( e
o g x
e
o g y
e
o g z
) = ( e
o g x
1 1 ) ( 1 e
o g y
1 ) ( 1 1 e
o g z
) =
= ( l o g x ( e 1 1 ) ) ( l o g y ( 1 e 1 ) ) ( l o g z ( 1 1 e ) )
l u e g o l o s v e c t o r e s ( e 1 1 ) ( 1 e 1 ) ( 1 1 e ) 2 R
0
R
0
R
0
f o r m a n u n s i s t e m a d e
g e n e r a d o r e s .
C l a r a m e n t e s o n i n d e p e n d i e n t e s , v e a m o s :
d e (
1
( e 1 1 ) ) (
2
( 1 e 1 ) ) (
3
( 1 1 e ) ) = ( 1 1 1 )
t e n e m o s
( e
1
e
2
e
3
) = ( 1 1 1 ) ) e
= 1 8 i = 1 2 3 )
i
= 0 8 i = 1 2 3
p o r l o q u e f o r m a n u n a b a s e d e d i c h o e s p a c i o v e c t o r i a l .
2 D e m o s t r a r q u e e l c o n j u n t o E d e l a s s u c e s i o n e s n u m e r i c a s
u = ( u
1
u
2
u
n
) = ( u
n
) n 2 N
d e n u m e r o s r e a l e s p r o v i s t a d e d o s l e y e s d e c o m p o s i c i o n , u n a i n t e r n a + y u n a e x t e r n a
, d e n i d a s m e d i a n t e :
8 u v 2 E 8 2 R
u + v = ( u
n
+ v
n
) 8 n 2 N
u = ( u
n
) 8 n 2 N
e s u n e s p a c i o v e c t o r i a l .
S o l u c i o n :
P r i m e r o , p r o b a r e m o s q u e l a o p e r a c i o n ( i n t e r n a ) + d o t a a E d e e s t r u c t u r a d e g r u p o
a b e l i a n o
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2 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
A s o c i a t i v i d a d
u + ( v + w ) = ( u
n
+ ( v + w )
n
) = ( u
n
+ ( v
n
+ w
n
) ) =
( 1 )
= ( ( u
n
+ v
n
) + w
n
) = ( ( u + v )
n
+ w
n
) = ( u + v ) + w
( 1 ) R t i e n e e s t r u c t u r a d e g r u p o , c o n l a o p e r a c i o n +
C o n m u t a t i v i d a d
( u + v ) = ( u
n
+ v
n
) = ( v
n
+ v
n
) = ( v + u )
E x i s t e n c i a d e e l e m e n t o n e u t r o
v e a m o s q u e e x i s t e e 2 E t a l q u e u + e = u 8 u 2 E
s i u + e = ( u
n
+ e
n
) = u 8 u 2 E , e n t o n c e s u
n
+ e
n
= u
n
8 n 2 N , d e d o n d e
e
n
= 0 8 n 2 N y e = ( 0 0 : : : 0 : : : ) , l u e g o e e x i s t e y e s u n i c o .
E x i s t e n c i a d e e l e m e n t o s i m e t r i c o
h e m o s d e v e r q u e 8 u 2 E e x i s t e u
; 1
t a l q u e u + u
; 1
= e
s i u + u
; 1
= ( u
n
+ u
; 1
n
) = e , e n t o n c e s u
n
+ u
; 1
n
= 0 8 n 2 N , d e d o n d e u
; 1
n
=
; u
n
8 n 2 N y u
; 1
= ( ; u
n
) l u e g o u
; 1
e x i s t e y e s u n i c o .
V e a m o s a h o r a q u e l a o p e r a c i o n ( e x t e r n a ) v e r i c a l a s c u a t r o p r o p i e d a d e s , n e c e s a r i a s
p a r a q u e e l g r u p o a b e l i a n o E s e a u n R - e s p a c i o v e c t o r i a l
P r i m e r a l e y d i s t r i b u t i v a
8 u v 2 E 8 2 R
( u + v ) = ( ( u + v )
n
) = ( ( u
n
+ v
n
) ) = ( u
n
+ v
n
) =
= ( u
n
) + ( v
n
) = ( u
n
) + ( v
n
) = u + v
S e g u n d a l e y d i s t r i b u t i v a
8 2 R 8 u 2 E
( + ) u = ( ( + ) u
n
) = ( u
n
+ u
n
) = ( u
n
) + ( u
n
) =
= ( u
n
) + ( u
n
) = u + u
A s o c i a t i v i d a d d e l o s e s c a l a r e s
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E s p a c i o s v e c t o r i a l e s 2 7
8 2 R 8 u 2 E
( ) u = ( ( ) u
n
) = ( ( u
n
) ) = ( u
n
) = ( ( u
n
) ) =
= ( u )
P r o p i e d a d d e l e l e m e n t o u n i d a d d e l c u e r p o
S e a 1 2 R y 8 u 2 E
1 u = ( 1 u
n
) = ( u
n
) = u
l u e g o E e s u n R - e s p a c i o v e c t o r i a l .
3 S e a F ( R R ) e l e s p a c i o v e c t o r i a l d e t o d a s l a s f u n c i o n e s d e R e n R . E s t u -
d i a r , p a r a q u e v a l o r e s d e k 2 R
W = f f 2 F ( R R ) = f ( 1 ) = k g
e s u n s u b e s p a c i o v e c t o r i a l d e F
S o l u c i o n :
R e c o r d e m o s q u e F e s u n s u b e s p a c i o v e c t o r i a l d e l K - e s p a c i o v e c t o r i a l E s i y s o l a m e n t e
s i :
8 2 K 8 u v 2 F e n t o n c e s u + v 2 F
S e a n p u e s 2 R y f g 2 F ( R R )
( f + g ) 2 F ( R R ) s i y s o l o s i ( f + g ) ( 1 ) = k
c o m p r o b e m o s s i e s t o e s a s
( f + g ) ( 1 ) = ( f ) ( 1 ) + ( g ) ( 1 ) = f ( 1 ) + g ( 1 ) = k + k = ( + ) k
l u e g o ( + ) k = k 8 2 R s i y s o l o s i k = 0 , p o r l o t a n t o W e s s u b e s p a c i o
v e c t o r i a l s i y s o l o s i k = 0
4 S e a f e
1
e
2
e
3
g u n a b a s e d e l R - e s p a c i o v e c t o r i a l R
3
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2 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
> D e t e r m i n a n l o s v e c t o r e s a e
1
+ b e
2
c e
2
+ d e
3
e e
3
+ f e
1
, c o n a b c d e f e s c a l a r e s
n o n u l o s , u n a b a s e d e E ?
A p l i c a r e l r e s u l t a d o a l a s f a m i l i a s d e v e c t o r e s
a ) ( 1 1 0 ) ( 0 1 1 ) ( 1 0 ; 1 )
b ) ( 3 1 0 ) ( 0 2 1 ) ( 1 0 2 )
r e f e r i d o s a l a b a s e n a t u r a l d e R
3
S o l u c i o n :
P u e s t o q u e e l n u m e r o d e v e c t o r e s d a d o c o i n c i d e c o n l a d i m e n s i o n d e l e s p a c i o , e s t o s
v e c t o r e s f o r m a n b a s e s i y s o l o s i s o n i n d e p e n d i e n t e s . R e c o r d e m o s q u e u n a c o l e c c i o n d e
v e c t o r e s f e
1
: : : e
n
g d e u n K - e s p a c i o v e c t o r i a l s o n i n d e p e n d i e n t e s s i y s o l o s i :
1
e
1
+ +
n
e
n
= 0 ,
1
= =
n
= 0
V e a m o s p u e s ,
1
( a e
1
+ b e
2
) +
2
( c e
2
+ d e
3
) +
3
( e e
3
+ f e
1
) = 0
(
1
a +
3
f ) e
1
+ (
1
b +
2
c ) e
2
+ (
2
d +
3
e ) e
3
= 0
Y p u e s t o q u e f e
1
e
2
e
3
g e s b a s e , t e n e m o s
1
a +
3
f = 0
1
b +
2
c = 0
2
d +
3
e = 0
9
>
>
=
>
>
,
1
a b +
3
f b = 0
1
a b +
2
a c = 0
2
d +
3
e = 0
9
>
>
=
>
>
)
3
f b ;
2
a c = 0
2
d +
3
e = 0
)
,
3
f b d ;
2
a c d = 0
2
a c d +
3
a c e = 0
)
)
3
( f b d + a c e ) = 0
L u e g o , s i f b d + a c e 6= 0 )
3
= 0
2
= 0
1
= 0 y l o s v e c t o r e s s e r a n i n d e p e n d i e n t e s
y f o r m a r a n b a s e ( s i f b d + a c e = 0 c u a l q u i e r
3
2 R e s s o l u c i o n d e l s i s t e m a y p o r l o
t a n t o , l o s v e c t o r e s d a d o s , n o p u e d e n f o r m a r b a s e . ) .
A p l i c a n d o e l r e s u l t a d o a l a s f a m i l i a s d a d a s , t e n e m o s
a ) ( 1 1 0 ) = ( 1 0 0 ) + ( 0 1 0 ) ) a = b = 1
( 0 1 1 ) = ( 0 1 0 ) + ( 0 0 1 ) ) c = d = 1
( 1 0 ; 1 ) = ( 1 0 0 ) ; ( 0 0 1 ) ) e = 1 f = ; 1
9
>
=
>
) f b d = ; a c e
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E s p a c i o s v e c t o r i a l e s 2 9
l u e g o , s o n d e p e n d i e n t e s ( l a r e l a c i o n d e d e p e n d e n c i a e s ( 1 1 0 ) ; ( 0 1 1 ) = ( 1 0 1 )
b ) ( 3 1 0 ) = 3 ( 1 0 0 ) + ( 0 1 0 ) ) a = 3 b = 1
( 0 2 1 ) = 2 ( 0 1 0 ) + ( 0 0 1 ) ) c = 2 d = 1
( 1 0 2 ) = ( 1 0 0 ) + 2 ( 0 0 1 ) ) e = 1 f = 2
9
>
=
>
) f b d 6= ; a c e
l u e g o , s o n i n d e p e n d i e n t e s , y p o r l o t a n t o f o r m a n b a s e .
5 S e a E u n e s p a c i o v e c t o r i a l s o b r e C d e d i m e n s i o n n y s e a f u
i
g
1 i n
u n a b a s e .
P o r r e s t r i c c i o n d e l c u e r p o d e e s c a l a r e s , E p u e d e c o n s i d e r a r s e c o m o u n e s p a c i o v e c t o r i a l
s o b r e R
D e m o s t r a r q u e l o s 2 n v e c t o r e s f u
1
: : : u
n
i u
1
: : : i u
n
g f o r m a n u n a b a s e d e E s o b r e
R . D e d u c i r d e a q u q u e d i m E
R
= 2 d i m E
C
N o t a : h e m o s l l a m a d o E
C
E
R
a E c o m o C e s p a c i o v e c t o r i a l y c o m o R e s p a c i o
v e c t o r i a l r e s p e c t i v a m e n t e .
S o l u c i o n :
A n t e t o d o , n o t a m o s q u e l o s v e c t o r e s d e E
C
y E
R
s o n l o s m i s m o s . V e a m o s p r i m e r o q u e
l o s v e c t o r e s d a d o s s o n i n d e p e n d i e n t e s e n E
R
c o n s i d e r e m o s u n a c o m b i n a c i o n l i n e a l
i g u a l a d a a c e r o :
1
u
1
+ +
n
u
n
+
n + 1
i u
1
+ +
2 n
i u
n
= 0 c o n
j
2 R j = 1 : : : 2 n
s u m e r g i e n d o E
R
e n E
C
e s t a i g u a l d a d p u e d e e s c r i b i r s e
(
1
+
n + 1
i ) u
1
+ + (
n
+
2 n
i ) u
n
= 0 c o n
j
+
n + j
i 2 C
y p u e s t o q u e f u
i
g s o n b a s e d e E
C
, t e n e m o s
j
+
n + j
i = 0 8 j = 1 : : : n
p o r l o q u e :
j
=
n + j
= 0 8 j = 1 : : : n
y p o r l o t a n t o , l o s v e c t o r e s f u
1
: : : u
n
i u
1
: : : i u
n
g s o n i n d e p e n d i e n t e s . V e a m o s
a h o r a q u e g e n e r a n E
R
S i u 2 E
R
, e n t o n c e s u 2 E
C
y p o r l o t a n t o
u =
1
u
1
+ +
n
u
n
c o n j 2 C j = 1 : : : n
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3 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
e s d e c i r ,
j
= a
j
+ b
j
i j = 1 : : : n c o n a
j
b
j
2 R
l u e g o
u = ( a
1
+ b
1
i ) u
1
+ + ( a
n
+ b
n
i ) u
n
=
= a
1
u
1
+ b
1
i ) u
1
+ + a
n
u
n
+ b
n
i u
n
=
= a
1
u
1
+ + a
n
u
n
+ b
1
i u
1
+ + b
n
i u
n
l u e g o , s o n t a m b i e n g e n e r a d o r e s . P o r s e r u n s i s t e m a d e g e n e r a d o r e s i n d e p e n d i e n t e s s o n
b a s e , y p o r l o t a n t o
d i m E
R
= 2 : d i m E
C
6 S e a E = R
3
. D e c i r s i l o s v e c t o r e s f ( 1 2 3 ) ( 2 5 8 ) ( 1 3 7 ) g s o n d e p e n d i e n t e s
o i n d e p e n d i e n t e s .
S o l u c i o n :
E l m e t o d o q u e v a m o s a u s a r a q u p a r a l a d i s c u s i o n d e l a d e p e n d e n c i a o i n d e p e n d e n c i a
s e a p o y a e n l a s p r o p o s i c i o n e s s i g u i e n t e s .
a ) D a d o s p v e c t o r e s , p n , d e u n e s p a c i o v e c t o r i a l d e d i m e n s i o n n , x
i
=
( a
1
i
: : : a
n
i
) 1 i p , s i l o s c o e c i e n t e s a
j
i
s o n n u l o s p a r a i > j c o n a
i
i
6= 0 ( e s
d e c i r , s i c o l o c a m o s l o s v e c t o r e s e n c o l u m n a , l a m a t r i z o b t e n i d a e s t a l q u e p o r e n c i m a
d e l a d i a g o n a l p r i n c i p a l , l o s e l e m e n t o s s o n t o d o s n u l o s ) , e n t o n c e s l o s v e c t o r e s s o n
i n d e p e n d i e n t e s ( e s u n a c o n d i c i o n s u c i e n t e , p e r o n o n e c e s a r i a ) . A n a l o g a m e n t e , s i l o s
c o e c i e n t e s a
j
i
s o n n u l o s p a r a i < j c o n a
i
i
6= 0 ( e s d e c i r , s i c o l o c a m o s l o s v e c t o r e s
e n c o l u m n a , l a m a t r i z o b t e n i d a e s t a l q u e p o r d e b a j o d e l a d i a g o n a l p r i n c i p a l , l o s
e l e m e n t o s s o n t o d o s n u l o s ) , t a m b i e n s o n i n d e p e n d i e n t e s .
b ) E l r a n g o d e u n s i s t e m a d e v e c t o r e s n o v a r a s i a u n o d e e l l o s l e s u m a m o s u n a
c o m b i n a c i o n l i n e a l d e l o s d e m a s , p o r l o t a n t o p a r a i n v e s t i g a r l a s d e p e n d e n c i a o n o d e
l o s v e c t o r e s d a d o s l o s c o l o c a r e m o s e n c o l u m n a y u x t a p o n i e n d o l o s y h a r e m o s o p e r a c i o n e s
e l e m e n t a l e s d e l a o c o l u m n a p a r a c o n s e g u i r l o s c e r o s n e c e s a r i o s p a r a c o n o c e r e l r a n g o
d e l a m a t r i z , e s d e c i r l a d i m e n s i o n d e l s u b e s p a c i o q u e e n g e n d r a n
x
1
x
2
x
3
x
0
1
x
0
2
x
0
3
x
0 0
1
x
0 0
2
x
0 0
3
0
@
1 2 1
2 5 3
3 8 8
1
A
0
@
1 0 0
2 1 1
3 2 5
1
A
0
@
1 0 0
2 1 0
3 2 3
1
A
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E s p a c i o s v e c t o r i a l e s 3 1
x
0
1
= x
1
x
0
2
= ; 2 x
1
+ x
2
x
0
3
= ; x
1
+ x
3
x
0 0
1
= x
0
1
x
0 0
2
= x
0
2
x
0 0
3
= ; x
0
2
+ x
0
3
L o s t r e s v e c t o r e s c u m p l e n l a c o n d i c i o n e s t a b l e c i d a e n a ) , l u e g o s o n i n d e p e n d i e n t e s .
7 H a l l a r 2 R p a r a q u e ( ; 3 7 ; 6 ) 2 R
4
p e r t e n e z c a a l s u b e s p a c i o F R
g e n e r a d o p o r ( 1 2 ; 5 3 ) y ( 2 ; 1 4 7 )
S o l u c i o n :
P a r a q u e e l v e c t o r ( ; 3 7 ; 6 ) p e r t e n e z c a a F e s c o n d i c i o n n e c e s a r i a y s u c i e n t e
q u e p u e d a p o n e r s e e n c o m b i n a c i o n l i n e a l d e l o s g e n e r a d o r e s d e F :
( ; 3 7 ; 6 ) = a ( 1 2 ; 5 3 ) + b ( 2 ; 1 4 7 )
o b l i g a n d o p u e s a l a c o m p a t i b i l i d a d d e l s i s t e m a r e s u l t a n t e
= a + 2 b
= 2 a ; b
; 3 7 = ; 5 a + 4 b
; 6 = 3 a + 7 b
9
>
>
>
=
>
>
>
a = 5 b = ; 3 = ; 1 = 1 3
l u e g o e l v e c t o r ( ; 3 7 ; 6 ) 2 F s i y s o l o s i = ; 1 y = 1 3 .
8 S e a E = R
2
y W e l s u b e s p a c i o e n g e n d r a d o p o r e l v e c t o r ( 1 1 ) . S i U e s e l
s u b e s p a c i o e n g e n d r a d o p o r e l v e c t o r ( 0 1 ) . P r o b a r q u e E e s s u m a d i r e c t a d e W y
U . S e a a h o r a U
0
e l s u b e s p a c i o e n g e n d r a d o p o r e l v e c t o r ( 3 1 ) . P r o b a r q u e t a m b i e n
s e v e r i c a E = W U
0
( n o u n i c i d a d d e l c o m p l e m e n t a r i o ) .
S o l u c i o n :
R e c o r d e m o s q u e s i F G s o n s u b e s p a c i o s d e E e s t o s f o r m a n s u m a d i r e c t a s i y s o l o
s i
F \ G = f 0 g
S i F y G f o r m a n s u m a d i r e c t a y a d e m a s s e v e r i c a q u e
F + G = E
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3 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
d i r e m o s q u e E e s s u m a d i r e c t a d e e s t o s d o s s u b e s p a c i o s y l o n o t a r e m o s p o r F G
S i E e s d e d i m e n s i o n n i t a y F y G f o r m a n s u m a d i r e c t a , p a r a q u e F + G = E
b a s t a c o m p r o b a r q u e
d i m F + d i m G = d i m E
T o m e m o s p u e s x 2 W \ U , e s x = ( 1 1 ) p o r s e r d e W , y x = ( 0 1 ) p o r s e r d e
U
I d e n t i c a n d o ( 1 1 ) = ( 0 1 ) e s
= 0 = 0
l u e g o x = 0 y p o r t a n t o l a s u m a e s d i r e c t a . P u e s t o q u e d i m E = 2 y d i m W =
d i m U = 1 s e t i e n e
d i m W + d i m U = d i m E
l u e g o , e n e f e c t o , E = W U
S e a a h o r a y 2 W \ U
0
c o m o a n t e s : y = ( 1 1 ) = ( 3 1 ) , d e d o n d e
= 3 =
y d e a q u s e d e d u c e = = 0 , e s d e c i r , y = 0 l u e g o W y U
0
f o r m a n t a m b i e n
s u m a d i r e c t a y d i m W + d i m U
0
= 2 = d i m E , p o r t a n t o e s t a m b i e n E = W U
0
9 S e a P
3
e l e s p a c i o v e c t o r i a l d e p o l i n o m i o s d e u n a v a r i a b l e d e g r a d o i n f e r i o r o i g u a l
a 3 a c o e c i e n t e s e n R
a ) P r o b a r q u e l o s p o l i n o m i o s p 2 P
3
q u e v e r i c a n p ( 1 ) = p
0
( 1 ) = 0 ( s i e n d o p
0
e l
p o l i n o m i o d e r i v a d o d e p ) f o r m a n u n s u b e s p a c i o v e c t o r i a l F d e P
3
. D a r s u d i m e n s i o n
b ) L o s p o l i n o m i o s ( x ; 1 )
2
y x ( x ; 1 )
2
, > s o n l i n e a l m e n t e i n d e p e n d i e n t e s ? D a r u n a
b a s e d e F
c ) E n c o n t r a r d o s p o l i n o m i o s p a r a c o m p l e t a r l a b a s e d e F y f o r m a r u n a b a s e d e P
3
D e t e r m i n a r u n s u b e s p a c i o v e c t o r i a l c o m p l e m e n t a r i o E d e F e n P
3
S o l u c i o n :
a ) S e a n p q 2 F v e a m o s s i p ; q 2 F 8 2 R
( p ; q ) ( 1 ) = p ( 1 ) ; q ( 1 ) = 0 ; 0 = 0
( p ; q )
0
( 1 ) = ( p
0
; q
0
) ( 1 ) = p
0
( 1 ) ; q
0
( 1 ) = 0 ; 0 = 0
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E s p a c i o s v e c t o r i a l e s 3 3
L u e g o , e n e f e c t o , p ; q 2 F , y F e s p u e s u n s u b e s p a c i o v e c t o r i a l .
S e a p ( x ) = a
3
x
3
+ a
2
x
2
+ a
1
x + a
0
2 F , l u e g o
p ( 1 ) = a
3
+ a
2
+ a
1
+ a
0
= 0
p
0
( 1 ) = 3 a
3
+ 2 a
2
+ a
1
= 0
)
a
1
= ; 3 a
3
; 2 a
2
a
0
= 2 a
3
+ a
2
y p o r t a n t o ,
p ( x ) = a
3
x
3
+ a
2
x
2
+ ( ; 3 a
3
; 2 a
2
) x + ( 2 a
3
+ a
2
) =
= a
3
( x
3
; 3 x + 2 ) + a
2
( x
2
; 2 x + 1 ) = a
3
p
1
( x ) + a
2
p
2
( x )
y p
1
( x ) p
2
( x ) 2 F ( p
1
( 1 ) = p
0
1
( 1 ) = 0 y p
2
( 1 ) = p
0
2
( 1 ) = 0 ) p o r l o q u e s o n
g e n e r a d o r e s .
Y s o n i n d e p e n d i e n t e s , p u e s d e
a
3
p
1
( x ) + a
2
p
2
( x ) = 0 s e t i e n e a
3
= a
2
= 0
l u e g o s o n b a s e , y d i m F = 2
O t r a f o r m a :
D e h e c h o , b a s t a o b s e r v a r q u e s i p ( x ) 2 F e n t o n c e s p ( x ) = ( x ; 1 )
2
( a x + b ) = a x ( x ;
1 )
2
+ b ( x ; 1 )
2
= a p
1
( x ) + b p
2
( x ) l u e g o F e s e l c o n j u n t o d e p o l i n o m i o s g e n e r a d o p o r
p
1
( x ) p
2
( x ) c o n p
1
( x ) y p
2
( x ) i n d e p e n d i e n t e s , p o r l o q u e F e s u n s u b e s p a c i o v e c t o r i a l
d e d i m e n s i o n 2 y e s t o s d o s p o l i n o m i o s d e t e r m i n a n u n a b a s e
b ) S e a ( x ; 1 )
2
+ x ( x ; 1 )
2
= 0 , r e o r d e n a n d o t e r m i n o s t e n e m o s
x
3
+ ( ; 2 ) x
2
+ ( ; 2 ) x + = 0
y p o r t a n t o , = = 0 , e s d e c i r , s o n i n d e p e n d i e n t e s . E n a h e m o s o b s e r v a d o , q u e
( x ; 1 )
2
x ( x ; 1 )
2
2 F ( p u e s F e s e l c o n j u n t o d e p o l i n o m i o s d e g r a d o m e n o r o i g u a l
q u e t r e s t a l e s q u e t i e n e n a 1 c o m o r a z d e m u l t i p l i c i d a d p o r l o m e n o s d o s ) , l u e g o , s o n
b a s e d e F
c ) L o s v e c t o r e s x
2
; 2 x + 1 x
3
; 2 x
2
+ x s o n i n d e p e n d i e n t e s y a q u e f o r m a n u n a b a s e
d e F 1 x s o n v e c t o r e s i n d e p e n d i e n t e s d e x
2
; 2 x + 1 x
3
; 2 x
2
+ x , y a q u e
1 + x + ( x
2
; 2 x + 1 ) + ( x
3
; 2 x
2
+ x ) = 0 )
x
3
+ ( ; 2 ) x
2
+ ( ; 2 + ) x + + = 0
d e d o n d e = = = = 0
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3 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
l u e g o G = 1 x ] e s u n s u b e s p a c i o c o m p l e m e n t a r i o d e F
1 0 S e a n A = f a
1
a
2
a
3
g , B = f b
1
b
2
b
3
g d o s b a s e s d e l e s p a c i o v e c t o r i a l R
3
r e l a c i o n a d a
m e d i a n t e :
8
>
<
>
:
a
1
= b
1
; 3 b
2
+ 4 b
3
a
2
= b
2
+ b
3
a
3
= b
1
+ b
2
+ b
3
a ) H a l l a r l a m a t r i z q u e t r a n s f o r m a l a s c o o r d e n a d a s d e l o s v e c t o r e s d e l a b a s e B a l a
A
b ) S e a C = f c
1
c
2
c
3
g u n a n u e v a b a s e c u y a s c o o r d e n a d a s r e s p e c t o d e B s o n :
8
>
<
>
:
c
1
= b
1
; b
2
+ b
3
c
2
= ; b
1
+ b
2
c
3
= b
2
; b
3
H a l l a r l a m a t r i z d e t r a n s f o r m a c i o n d e B a C y d e A a C
S o l u c i o n :
a ) R e c o r d e m o s q u e l a m a t r i z S d e p a s o d e A a B e s l a m a t r i z c u a d r a d a c u y a s c o l u m -
n a s s o n l a s c o o r d e n a d a s d e l o s v e c t o r e s d e A e x p r e s a d o s e n l a b a s e B . L u e g o :
S =
0
@
1 0 1
; 3 1 1
4 1 1
1
A
m a t r i z d e p a s o d e A a B
y e s t a m a t r i z e s t a l q u e s i c o m p o n e m o s d i c h a m a t r i z c o n u n v e c t o r c o l u m n a c u y o s
c o m p o n e n t e s s o n l a s c o o r d e n a d a s d e u n v e c t o r d e R
3
e n l a b a s e A , e l r e s u l t a d o e s
e l m i s m o v e c t o r ( v e c t o r c o l u m n a ) c u y o s c o m p o n e n t e s s o n l a s c o o r d e n a d a s d e l v e c t o r ,
p e r o e x p r e s a d o e n l a b a s e B
O b v i a m e n t e , l a m a t r i z d e p a s o d e B a A s e r a
S
; 1
=
1
7
0
@
0 ; 1 1
; 7 3 4
7 1 ; 1
1
A
( o b s e r v e s e q u e n e c e s i t a m o s l a e x p r e s i o n d e l o s v e c t o r e s d e l a b a s e B e n f u n c i o n d e l a
b a s e A , p o r l o q u e t e n e m o s q u e i n v e r t i r e l s i s t e m a d a d o ) .
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E s p a c i o s v e c t o r i a l e s 3 5
b ) A n a l o g a m e n t e , l a m a t r i z d e p a s o d e C a B e s
T =
0
@
1 ; 1 0
; 1 1 1
1 0 ; 1
1
A
l u e g o , l a m a t r i z d e p a s o d e B a C e s
T
; 1
=
0
@
1 1 1
0 1 1
1 1 0
1
A
y s i c o m p o n e m o s l a s m a t r i c e s S y T
; 1
T
; 1
S =
0
@
2 2 3
1 2 2
; 2 1 2
1
A
n o s p r o p o r c i o n a , o b v i a m e n t e , l a m a t r i z d e p a s o d e A a C
1 1 E s t u d i a r s i l o s v e c t o r e s w
1
= ( 0 1 ; 2 1 ) w
2
= ( 1 1 2 ; 1 ) w
3
= ( 1 0 0 1 )
w
4
= ( 2 2 0 ; 1 ) f o r m a n o n o , u n a b a s e d e R
4
S o l u c i o n :
P a r a q u e f o r m e n b a s e e s c o n d i c i o n n e c e s a r i a y s u c i e n t e q u e s e a n l i n e a l m e n t e i n d e p e n -
d i e n t e s , e s d e c i r ,
1
w
1
+
2
w
2
+
3
w
3
+
4
w
4
= 0 ,
1
=
2
=
3
=
4
= 0
l o q u e e q u i v a l e a d e c i r , q u e e l s i s t e m a
0
1
+ 1
2
+ 1
3
+ 2
4
= 0
1
1
+ 1
2
+ 0
3
+ 2
4
= 0
; 2
1
+ 2
2
+ 0
3
+ 0
4
= 0
1
1
; 1
2
+ 1
3
; 1
4
= 0
9
>
>
>
=
>
>
>
t e n g a s o l u c i o n u n i c a l o q u e e q u i v a l e a q u e
D =
0 1 1 2
1 1 0 2
; 2 2 0 0
1 ; 1 1 ; 1
6= 0
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3 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
D = ; 8 6= 0 , l u e g o , e n e f e c t o , s o n b a s e .
O b s e r v a m o s q u e , p a r a v e r s i n v e c t o r e s d e u n e s p a c i o v e c t o r i a l d e d i m e n s i o n n , f o r m a n
b a s e b a s t a c a l c u l a r e l d e t e r m i n a n t e d e l a m a t r i z , f o r m a d a p o r l o s v e c t o r e s c o l u m n a q u e
e x p r e s a n l o s v e c t o r e s d a d o s r e s p e c t o u n a b a s e y v e r s i e s o n o d i s t i n t o d e c e r o .
1 2 E n e l e s p a c i o v e c t o r i a l R
4
s e c o n s i d e r a n l o s s u b e s p a c i o s E
1
y E
2
g e n e r a d o s
p o r l o s v e c t o r e s ( 1 , 1 , 1 , 1 ) y ( 1 , - 1 , 1 , - 1 ) p a r a E
1
, y ( 1 , 2 , 0 , 1 ) , ( 1 , 2 , 1 , 2 ) y ( 3 , 1 , 3 , 1 ) p a r a
E
2
H a l l a r l a s d i m e n s i o n e s d e l s u b e s p a c i o i n t e r s e c c i o n y d e l s u b e s p a c i o s u m a .
S o l u c i o n :
O b s e r v a m o s q u e d i m E
1
= 2 y a q u e ( 1 , 1 , 1 , 1 ) , ( 1 , - 1 , 1 , - 1 ) s o n i n d e p e n d i e n t e s .
V e a m o s c u a l e s e l s u b e s p a c i o E
1
\ E
2
: s i v 2 E
1
\ E
2
, e n t o n c e s
v =
1
( 1 1 1 1 ) +
2
( 1 ; 1 1 ; 1 ) =
=
1
( 1 2 0 1 ) +
2
( 1 2 1 2 ) +
3
( 3 1 3 1 )
e s d e c i r ,
1
+
2
=
1
+
2
+ 3
3
1
;
2
= 2
1
+ 2
2
+
3
1
+
2
=
2
+ 3
3
1
;
2
=
1
+ 2
2
+
3
9
>
>
>
=
>
>
>
)
1
= 0
2
1
= 3
2
+ 4
3
2
2
= ;
2
+ 2
3
9
>
=
>
p o r l o q u e , d a n d o v a l o r e s c u a l e s q u i e r a a l o s e s c a l a r e s
2
3
, o b t e n d r e m o s l o s v e c t o r e s
d e E
1
\ E
2
, y p u e s t o q u e h a y d o s p a r a m e t r o s l i b r e s d i m E
1
\ E
2
= 2
P o r e j e m p l o , p a r a
2
= ; 1
3
= 1 , s e t i e n e
w
1
= ( 3 1 3 1 ) ; ( 1 2 1 2 ) = ( 2 ; 1 2 ; 1 ) 2 E
1
\ E
2
p a r a
2
=
3
= 1
w
2
= ( 1 2 1 2 ) + ( 3 1 3 1 ) = ( 4 3 4 3 ) 2 E
1
\ E
2
o b s e r v a m o s q u e w
1
y w
2
s o n i n d e p e n d i e n t e s p o r l o q u e d i m E
1
\ E
2
2 y p u e s t o
q u e
E
1
\ E
2
E
1
y d i m E
1
= 2
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E s p a c i o s v e c t o r i a l e s 3 7
s e t i e n e q u e E
1
\ E
2
= E
1
y d i m E
1
\ E
2
= 2
S a b e m o s q u e d i m ( E
1
+ E
2
) = d i m E
1
+ d i m E
2
; d i m ( E
1
\ E
2
) , l u e g o d i m ( E
1
+ E
2
) =
d i m E
2
( T e n e m o s q u e E
1
+ E
2
= E
2
, p u e s E
2
E
1
+ E
2
y t i e n e n l a m i s m a d i m e n s i o n ) .
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S i s t e m a s d e e c u a c i o n e s l i n e a l e s . M a t r i c e s 3 9
C a p t u l o 3 S i s t e m a s d e e c u a c i o n e s l i n e a l e s . M a t r i c e s
1 D a d a l a m a t r i z
B =
0
@
8 ; 5 ; 1 3
8 ; 4 ; 2 9
4 ; 1 ; 6
1
A
r e d u c i r l a a u n a m a t r i z d i a g o n a l m e d i a n t e t r a n s f o r m a c i o n e s e l e m e n t a l e s d e l a s y c o l u m -
n a s .
S o l u c i o n :
T o m a m o s l a m a t r i z B y r e s t a m o s l a p r i m e r a l a a l a s e g u n d a y l a p r i m e r a a l d o b l e
d e l a t e r c e r a , q u e d a n d o
B
0
@
8 ; 5 ; 1 3
0 1 ; 1 6
0 3 1
1
A
= B
1
T o m a m o s a h o r a l a m a t r i z B
1
y r e s t a m o s a l a t e r c e r a l a e l t r i p l e d e l a s e g u n d a ,
q u e d a n d o
B
1
0
@
8 ; 5 ; 1 3
0 1 ; 1 6
0 0 4 9
1
A
= B
2
S o b r e B
2
, d i v i d i m o s l a t e r c e r a l a p o r 4 9
B
2
0
@
8 ; 5 ; 1 3
0 1 ; 1 6
0 0 1
1
A
= B
3
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4 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S o b r e B
3
, s u m a m o s a l a s e g u n d a l a d i e c i s e i s v e c e s l a t e r c e r a y a l a p r i m e r a t r e c e
v e c e s l a t e r c e r a
B
3
0
@
8 ; 5 0
0 1 0
0 0 1
1
A
= B
4
S o b r e B
4
, s u m a m o s a l a p r i m e r a l a c i n c o v e c e s l a s e g u n d a y t e n e m o s
B
4
0
@
8 0 0
0 1 0
0 0 1
1
A
= B
5
q u e e s y a d i a g o n a l p o d e m o s s e g u i r r e d u c i e n d o d i v i d i e n d o l a p r i m e r a l a p o r 8 ,
o b t e n i e n d o
B
5
0
@
1 0 0
0 1 0
0 0 1
1
A
= I
l u e g o B I
2 D e t e r m i n a r l a i n v e r s a d e l a m a t r i z
A =
0
@
1 4 8
0 1 2
; 1 2 3
1
A
d o n d e A 2 M
3
( R )
p o r e l m e t o d o d e l p i v o t e
S o l u c i o n :
Y u x t a p o n e m o s l a m a t r i z A y l a m a t r i z i d e n t i d a d I
0
B
B
@
1 4 8 . 1 0 0
0 1 2 . 0 1 0
; 1 2 3 . 0 0 1
1
C
C
A
= A
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S i s t e m a s d e e c u a c i o n e s l i n e a l e s . M a t r i c e s 4 1
y h a c e m o s l a s t r a n s f o r m a c i o n e s e l e m e n t a l e s d e l a s , n e c e s a r i a s p a r a c o n v e r t i r A e n
I . U n a v e z t e r m i n a d o e l p r o c e s o l a m a t r i z q u e a p a r e c e e n e l l u g a r q u e o c u p a b a I e s
l a m a t r i z A
; 1
i n v e r s a d e A
A
( a )
0
B
B
@
1 4 8 . 1 0 0
0 1 2 . 0 1 0
0 6 1 1 . 1 0 1
1
C
C
A
( b )
0
B
B
@
1 4 8 . 1 0 0
0 1 2 . 0 1 0
0 0 1 ; 1 6 ; 1
1
C
C
A
( c )
0
B
B
@
1 4 8 . 1 0 0
0 1 0 2 ; 1 1 2
0 0 1 ; 1 6 ; 1
1
C
C
A
( d )
0
B
B
@
1 4 0 9 ; 4 8 8
0 1 0 2 ; 1 1 2
0 0 1 ; 1 6 ; 1
1
C
C
A
( e )
0
B
B
@
1 0 0 1 ; 4 0
0 1 0 2 ; 1 1 2
0 0 1 ; 1 6 ; 1
1
C
C
A
y p o r c o n s i g u i e n t e
A
; 1
=
0
@
1 ; 4 0
2 ; 1 1 2
; 1 6 ; 1
1
A
( a ) A l a t e r c e r a l a d e A l e s u m a m o s l a p r i m e r a , o b t e n i e n d o A
1
( b ) A l a t e r c e r a l a d e A
1
m u l t i p l i c a d a p o r ; 1 l e s u m a m o s s e i s v e c e s l a s e g u n d a d e
A
1
o b t e n i e n d o A
2
( c ) A l a s e g u n d a l a d e A
2
l e r e s t a m o s d o s v e c e s l a t e r c e r a d e A
2
, o b t e n i e n d o A
3
( d ) A l a p r i m e r a l a d e A
3
l e r e s t a m o s o c h o v e c e s l a t e r c e r a , o b t e n i e n d o A
4
( e ) A l a p r i m e r a l a d e A
4
l e r e s t a m o s c u a t r o v e c e s l a s e g u n d a
3 R e s o l v e r e l s i s t e m a
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4 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
x ; 2 y + 3 z = 7
2 x + y ; 2 z = ; 2
3 x ; y + z = 6
9
>
=
>
S o l u c i o n :
E l s i s t e m a s e e x p r e s a e n f o r m a m a t r i c i a l p o r
0
@
1 ; 2 3
2 1 ; 2
3 ; 1 1
1
A
0
@
x
y
z
1
A
=
0
@
7
; 2
6
1
A
A X =
0
@
7
; 2
6
1
A
c o n X =
0
@
x
y
z
1
A
Y u x t a p o n e m o s a l a m a t r i z A l a m a t r i z c o l u m n a
0
@
7
; 2
6
1
A
, o b t e n i e n d o l a m a t r i z B y
h a c e m o s t r a n s f o r m a c i o n e s e l e m e n t a l e s a B p a r a p o d e r c o m p a r a r l o s r a n g o s d e A y
B
0
@
1 ; 2 3 7
2 1 ; 2 ; 2
3 ; 1 1 6
1
A
0
@
1 ; 2 3 7
0 5 ; 8 ; 1 6
0 5 ; 8 ; 1 5
1
A
0
@
1 ; 2 3 7
0 5 ; 8 ; 1 6
0 0 0 1
1
A
l u e g o r a n g o A = 2 , r a n g o B = 3 , y p o r t a n t o , e l s i s t e m a e s i n c o m p a t i b l e .
4 D i s c u t i r s e g u n l o s v a l o r e s d e a b c d e l s i s t e m a a c o e c i e n t e s e n K s i g u i e n t e
x + 2 y + 3 z + 4 t = a
2 x + 3 y + 4 z + t = b
3 x + 4 y + z + 2 t = c
4 x + y + 2 z + 3 t = d
9
>
>
>
=
>
>
>
s u p o n i e n d o : a ) K = Q b ) K = Z
S o l u c i o n :
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S i s t e m a s d e e c u a c i o n e s l i n e a l e s . M a t r i c e s 4 3
0
B
@
1 2 3 4 a
2 3 4 1 b
3 4 1 2 c
4 1 2 3 d
1
C
A
0
B
@
1 2 3 4 a
0 ; 1 ; 2 ; 7 ; 2 a + b
0 ; 2 ; 8 ; 1 0 ; 3 a + c
0 ; 7 ; 1 0 ; 1 3 ; 4 a + d
1
C
A
0
B
@
1 2 3 4 a
0 1 2 7 2 a ; b
0 0 4 ; 4 a ; 2 b + c
0 0 4 3 6 1 0 a ; 7 b + d
1
C
A
0
B
@
1 2 3 4 a
0 1 2 7 2 a ; b
0 0 4 ; 4 ; a + 2 b ; c
0 0 0 4 0 1 1 ; 9 b + c + d
1
C
A
0
B
@
1 0 2 0 3 0 4 0 1 0 a
0 4 0 8 0 2 8 0 8 0 a ; 4 0 b
0 0 4 0 0 a + 1 1 b ; 9 c + d
0 0 0 4 0 1 1 a ; 9 b + c + d
1
C
A
0
B
@
4 0 0 0 0 ; 9 a + b + c + 1 1 d
0 4 0 0 0 a + b + 1 1 c ; 9 d
0 0 4 0 0 a + 1 1 b ; 9 c + d
0 0 0 4 0 1 1 a ; 9 b + c + d
1
C
A
a ) p a r a K = Q l a s t r a n s f o r m a c i o n e s e l e m e n t a l e s r e a l i z a d a s s o n v a l i d a s e l s i s t e m a
t i e n e s o l u c i o n u n i c a
x =
; 9 a + b + c + 1 1 d
4 0
y =
a + b + 1 1 c ; 9 d
4 0
z =
a + 1 1 b ; 9 c + d
4 0
t =
1 1 a ; 9 b + c + d
4 0
b ) p a r a K = Z l a s t r a n s f o r m a c i o n e s e l e m e n t a l e s r e a l i z a d a s s o n v a l i d a s p a r a q u e h a y a
s o l u c i o n l o s e l e m e n t o s ; 9 a + b + c + 1 1 d a + b + 1 1 c ; 9 d a + 1 1 b ; 9 c + d 1 1 a ; 9 b + c + d
h a n d e s e r m u l t i p l o s d e 4 0 .
5 E s t u d i a r s e g u n l o s v a l o r e s d e a e l s i s t e m a
a x + y + z = a
x + a y + z = a
x + y + a z = a
9
>
=
>
r e s o l v i e n d o l o e n l o s c a s o s e n q u e e l l o s e a p o s i b l e .
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4 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S o l u c i o n :
H a l l e m o s e l v a l o r d e l d e t e r m i n a n t e d e l a m a t r i z a s o c i a d a a l s i s t e m a
d e t A = d e t
0
@
a 1 1
1 a 1
1 1 a
1
A
= ( a + 2 ) ( a ; 1 )
2
L u e g o , s i a 6= ; 2 y a 6= 1 e l s i s t e m a e s c o m p a t i b l e y d e t e r m i n a d o y p o r e l m e t o d o
d e C r a m e r t e n e m o s q u e l a s o l u c i o n e s
x =
a 1 1
a a 1
a 1 a
( a + 2 ) ( a ; 1 )
2
=
a
a + 2
y =
a a 1
1 a 1
1 a a
( a + 2 ) ( a ; 1 )
2
=
a
a + 2
z =
a 1 a
1 a a
1 1 a
( a + 2 ) ( a ; 1 )
2
=
a
a + 2
S i a = ; 2 r a n g o A = 2 y a q u e
; 2 1
1 ; 2
6= 0 , y r a n g o A = 3 s i e n d o A l a m a t r i z
o b t e n i d a d e A y u x t a p o n i e n d o l e l a m a t r i z c o l u m n a
0
@
a
a
a
1
A
r a n g o A = 3 y a q u e
; 2 1 ; 2
1 ; 2 ; 2
1 1 ; 2
6= 0
L u e g o e l s i s t e m a e s i n c o m p a t i b l e .
S i a = 1 r a n g o A = r a n g o A = 1 , l u e g o e l s i s t e m a e s c o m p a t i b l e i n d e t e r m i n a d o y e l
c o n j u n t o d e s o l u c i o n e s e s :
S = f ( x y z ) 2 R
3
x + y + z = 1 g
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S i s t e m a s d e e c u a c i o n e s l i n e a l e s . M a t r i c e s 4 5
6 R e s o l v e r e n M
2
( R ) e l s i s t e m a
2 x + 3 y + z =
5 5
; 5 1
x + 2 y ; 3 z =
; 1 3
; 3 ; 3
3 x + 5 y ; z =
5 8
; 8 ; 1
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
S o l u c i o n :
A
0
@
x
y
z
1
A
=
0
@
2 3 1
1 2 ; 3
3 5 ; 1
1
A
0
@
x
y
z
1
A
=
0
B
B
B
@
5 5
; 5 1
; 1 3
; 3 ; 3
5 8
; 8 ; 1
1
C
C
C
A
= B
T o m a m o s l a m a t r i z A d e l s i s t e m a a m p l i a d a c o n l a m a t r i z B y p r o c e d e m o s a e f e c t u a r
l a s o p o r t u n a s t r a n s f o r m a c i o n e s e l e m e n t a l e s
0
B
B
B
@
2 3 1
5 5
; 5 1
1 2 ; 3
; 1 3
; 3 ; 3
3 5 1
5 8
; 8 ; 1
1
C
C
C
A
0
B
B
B
@
1 2 ; 3
; 1 3
; 3 ; 3
2 3 1
5 5
; 5 1
3 5 1
5 8
; 8 ; 1
1
C
C
C
A
0
B
B
B
@
1 2 ; 3
; 1 3
; 3 ; 3
0 ; 1 7
7 ; 1
1 7
0 ; 1 8
8 ; 1
1 8
1
C
C
C
A
0
B
B
B
@
1 2 ; 3
; 1 3
; 3 ; 3
0 ; 1 7
7 ; 1
1 7
0 0 1
1 0
0 1
1
C
C
C
A
0
B
B
B
@
1 2 ; 3
; 1 3
; 3 ; 3
0 ; 1 0
0 ; 1
1 0
0 0 1
1 0
0 1
1
C
C
C
A
0
B
B
B
@
1 0 0
2 1
; 1 0
0 ; 1 0
0 ; 1
1 0
0 0 1
1 0
0 1
1
C
C
C
A
y p o r l o t a n t o l a s o l u c i o n d e l s i s t e m a e s
X =
2 1
; 1 0
Y =
0 1
; 1 0
Z =
1 0
0 1
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4 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
7 S e d i c e q u e A 2 M
3
( R ) e s m a g i c a s i a l s u m a r l o s e l e m e n t o s d e c a d a l a , d e c a d a
c o l u m n a , d e l a d i a g o n a l p r i n c i p a l , y d e l a d i a g o n a l s e c u n d a r i a , s e o b t i e n e s i e m p r e e l
m i s m o v a l o r . C o n s t r u i r t o d a s l a s m a t r i c e s m a g i c a s s i m e t r i c a s .
S o l u c i o n :
U n a m a t r i z A = ( a
i j
) e s s i m e t r i c a s i y s o l o s i a
i j
= a
j i
l u e g o l a s m a t r i c e s m a g i c a s
s i m e t r i c a s s e r a n d e l a f o r m a
0
@
x a b
a y c
b c z
1
A
c o n x + a + b = s a + y + c = s b + c + z = s x + y + z = s 2 b + y = s q u e
i n t e r p r e t a n d o l o c o m o u n s i s t e m a d e c i n c o e c u a c i o n e s c o n s i e t e i n c o g n i t a s
x + y + z ; s = 0
y + a + c ; s = 0
x + a + b ; s = 0
z + b + c ; s = 0
y + 2 b ; s = 0
9
>
>
>
>
>
=
>
>
>
>
>
r e s u l t a u n s i s t e m a h o m o g e n e o , p o r t a n t o c o m p a t i b l e , y q u e v a m o s a r e s o l v e r p o r t r a n s -
f o r m a c i o n e s e l e m e n t a l e s
0
B
B
B
@
1 1 1 0 0 0 ; 1
0 1 0 1 0 1 ; 1
1 0 0 1 1 0 ; 1
0 0 1 0 1 1 ; 1
0 1 0 0 2 0 ; 1
1
C
C
C
A
0
B
B
B
@
1 1 1 0 0 0 ; 1
0 1 0 1 0 1 ; 1
0 ; 1 ; 1 1 1 0 0
0 0 1 0 1 1 ; 1
0 0 0 ; 1 2 ; 1 0
1
C
C
C
A
0
B
B
B
@
1 1 1 0 0 0 ; 1
0 1 0 1 0 1 ; 1
0 0 ; 1 2 1 1 ; 1
0 0 1 0 1 1 ; 1
0 0 0 ; 1 2 ; 1 0
1
C
C
C
A
0
B
B
B
@
1 1 1 0 0 0 ; 1
0 0 1 1 0 1 ; 1
0 0 1 ; 2 ; 1 ; 1 1
0 0 0 2 2 2 ; 2
0 0 0 1 ; 2 1 0
1
C
C
C
A
0
B
B
B
@
1 1 1 0 0 0 ; 1
0 1 0 1 0 1 ; 1
0 0 1 ; 2 ; 1 ; 1 1
0 0 0 1 1 1 ; 1
0 0 0 0 ; 3 0 1
1
C
C
C
A
0
B
B
B
@
1 1 1 0 0 0 ; 1
0 1 0 1 0 1 ; 1
0 0 3 ; 6 0 ; 3 2
0 0 0 3 0 3 ; 2
0 0 0 0 ; 3 0 1
1
C
C
C
A
0
B
B
B
@
1 1 1 0 0 0 ; 1
0 3 0 0 0 0 ; 1
0 0 3 0 0 3 2
0 0 0 3 0 3 ; 2
0 0 0 0 ; 3 0 1
1
C
C
C
A
0
B
B
B
@
3 0 0 0 0 ; 3 0
0 3 0 0 0 0 ; 1
0 0 3 0 0 3 ; 2
0 0 0 3 0 3 ; 2
0 0 0 0 ; 3 0 1
1
C
C
C
A
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S i s t e m a s d e e c u a c i o n e s l i n e a l e s . M a t r i c e s 4 7
q u e d a n d o e l s i s t e m a d e r a n g o c i n c o :
3 x ; 3 c = 0
3 y ; s = 0
3 z + 3 c ; 2 s = 0
3 a + 3 c ; 2 s = 0
; 3 b + s = 0
9
>
>
>
>
>
=
>
>
>
>
>
,
3 x = 3 c
3 y = s
3 z = ; 3 c + 2 s
3 a = ; 3 c + 2 s
3 b = s
9
>
>
>
>
>
=
>
>
>
>
>
,
x = c
y =
s
3
= b
z =
2 s ; 3 c
3
= a
p o r l o t a n t o l a m a t r i z m a g i c a s i m e t r i c a b u s c a d a e s
0
B
B
B
@
c
2 s ; 3 c
3
s
3
2 s ; 3 c
3
s
3
c
s
3
c
2 s ; 3 c
3
1
C
C
C
A
= c
0
@
1 ; 1 0
; 1 0 1
0 1 ; 1
1
A
+
s
3
0
@
0 2 1
2 1 0
1 0 2
1
A
p a r a t o d o c s 2 R
8 D i s c u t i r y r e s o l v e r e n R e l s i s t e m a :
3 x + 2 y + 5 z = 1
4 x + 3 y + 6 z = 2
5 x + 4 y + 7 z = 3
6 x + 5 y + 8 z = 4
9
>
>
>
=
>
>
>
S o l u c i o n :
E s c r i t o m a t r i c i a l m e n t e , e l s i s t e m a e s
A
0
@
x
y
z
1
A
=
0
B
@
3 2 5
4 3 6
5 4 7
6 5 8
1
C
A
0
@
x
y
z
1
A
=
0
B
@
1
2
3
4
1
C
A
= B
T o m a m o s l a m a t r i z A o b t e n i d a d e A y u x t a p o n i e n d o l e l a m a t r i z c o l u m n a B y p r o -
c e d e m o s a e f e c t u a r t r a n s f o r m a c i o n e s e l e m e n t a l e s d e l a
0
B
@
3 2 5 1
4 3 6 2
5 4 7 3
6 5 8 4
1
C
A
0
B
@
3 2 5 1
0 ; 1 2 ; 2
0 ; 2 4 ; 4
0 ; 1 2 ; 2
1
C
A
0
B
@
3 2 5 1
0 ; 1 2 ; 2
0 0 0 0
0 0 0 0
1
C
A
0
B
@
6 9 0 1 2
0 ; 1 2 ; 2
0 0 0 0
0 0 0 0
1
C
A
0
B
@
2 3 0 4
0 ; 1 2 ; 2
0 0 0 0
0 0 0 0
1
C
A
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4 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
y t e n e m o s r a n g o A = r a n g o A = 2 < 3 , l u e g o e l s i s t e m a e s c o m p a t i b l e i n d e t e r m i n a d o
y e l c o n j u n t o d e s o l u c i o n e s e s
f ( x y z ) 2 R
3
= 2 x + 3 y = 4 ; y + 2 z = ; 2 g = f ( 2 ;
3
2
; 1 +
1
2
) 8 2 R g
9 D e t e r m i n a r X t a l q u e A X = B , s i e n d o
A =
0
@
1 1
1 0
1 1
1
A
y B =
0
@
2 1
0 2
2 1
1
A
S o l u c i o n :
T o m e m o s l a m a t r i z A
0
o b t e n i d a d e A y u x t a p o n i e n d o l e l a m a t r i z B y p r o c e d a m o s
a e f e c t u a r t r a n s f o r m a c i o n e s e l e m e n t a l e s d e l a
A
0
=
0
@
1 1 2 1
1 0 0 2
1 1 2 1
1
A
( a )
0
@
1 0 0 2
1 1 2 1
1 1 2 1
1
A
( b )
0
@
1 0 0 2
1 1 2 1
0 0 0 0
1
A
( c )
0
@
1 0 0 2
0 1 2 ; 1
0 0 0 0
1
A
Y t e n e m o s q u e r a n g A = 2 y r a n g A
0
= 2 p o r l o t a n t o e l s i s t e m a e s c o m p a t i b l e y
d e t e r m i n a d o y l a u n i c a s o l u c i o n e s :
X =
0 2
2 ; 1
( a ) P e r m u t a m o s l a p r i m e r a c o n l a s e g u n d a l a
( b ) A l a t e r c e r a l a l e r e s t a m o s l a s e g u n d a
( c ) A l a s e g u n d a l a l e r e s t a m o s l a p r i m e r a .
1 0 S e a n a b c d c u a t r o n u m e r o s r e a l e s e s t r i c t a m e n t e p o s i t i v o s . D e m o s t r a r q u e
e l s i s t e m a s i g u i e n t e n o p o s e e n i n g u n a s o l u c i o n e n R
x + y + z + t = a
x ; y ; z + t = b
; x ; y + z + t = c
; 3 x + y ; 3 z ; 7 t = d
9
>
>
>
=
>
>
>
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S i s t e m a s d e e c u a c i o n e s l i n e a l e s . M a t r i c e s 4 9
S o l u c i o n :
E l d e t e r m i n a n t e d e l s i s t e m a e s
1 1 1 1
1 ; 1 ; 1 1
; 1 ; 1 1 1
; 3 1 ; 3 ; 7
=
1 0 0 0
1 ; 2 0 0
; 1 0 ; 2 2
; 3 4 ; 4 ; 4
= 0
l u e g o e l s i s t e m a n o e s d e r a n g o m a x i m o
1 1 1
1 ; 1 ; 1
; 1 ; 1 1
=
1 0 0
1 ; 2 ; 2
; 1 0 2
= ; 4 6= 0
l u e g o e l s i s t e m a e s d e r a n g o t r e s y l a s t r e s p r i m e r a s e c u a c i o n e s s o n i n d e p e n d i e n t e s .
C o n s i d e r a m o s p u e s e l s i s t e m a
x + y + z = a ; t
x ; y ; z = b ; t
; x ; y + z = c ; t
9
>
=
>
q u e e s c o m p a t i b l e y d e t e r m i n a d o p o r s e r d e r a n g o m a x i m o y r e s o l v i e n d o p o r C r a m e r ,
t e n e m o s
x =
a + b
2
; t y = t ;
b + c
2
z =
a + c
2
; t
P a r a q u e e l s i s t e m a i n i c i a l s e a c o m p a t i b l e e s t o s v a l o r e s d e x y z h a l l a d o s , h a n d e
s a t i s f a c e r l a c u a r t a e c u a c i o n s u b s t i t u y e n d o p u e s , t e n e m o s
; 3 (
a + b
2
; t ) + ( t ;
b + c
2
) ; 3 (
a + c
2
; t ) ; 7 t = d
l u e g o l a c o m p a t i b i l i d a d i m p l i c a
; ( 3 a + 2 b + 2 c ) = d
y p u e s t o q u e a b c d s o n e s t r i c t a m e n t e p o s i t i v o s , e s t a i g u a l d a d e s i m p o s i b l e y e l
s i s t e m a e s i n c o m p a t i b l e .
1 1 R e s o l v e r e l s i g u i e n t e s i s t e m a d e e c u a c i o n e s e n C
x + y + z = a
x + w y + w
2
z = b
x + w
2
y + w z = b
9
>
=
>
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5 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
s a b i e n d o q u e w e s u n a r a z c u b i c a d e l a u n i d a d .
S o l u c i o n :
E l d e t e r m i n a n t e d e l s i s t e m a e s
1 1 1
1 w w
2
1 w
2
w
= ( w ; 1 )
2
; ( w
2
; 1 )
2
= 3 w ( w ; 1 )
( N o t a : p u e s t o q u e w
3
= 1 , s e t i e n e w
4
= w : : : y w
3
; 1 = ( w ; 1 ) ( w
2
+ w + 1 ) ) .
S i w ( w ; 1 ) 6= 0 , e l s i s t e m a e s c o m p a t i b l e y d e t e r m i n a d o p a r a t o d o a b 2 C y
r e s o l v i e n d o e l s i s t e m a p o r C r a m e r ,
x =
a 1 1
b w w
2
b w
2
w
3 w ( w ; 1 )
=
a + 2 b
3
y =
1 a 1
1 b w
2
1 b w
3 w ( w ; 1 )
=
a ; b
3
z =
1 1 a
1 w b
1 w
2
b
3 w ( w ; 1 )
=
a ; b
3
s i w ( w ; 1 ) = 0 , s e t i e n e q u e w = 0 o w ; 1 = 0 , p e r o s i w
3
= 1 , e s w 6= 0 ,
l u e g o h a d e s e r w ; 1 = 0 , e s d e c i r w = 1 , e n c u y o c a s o
0
@
1 1 1 a
1 1 1 b
1 1 1 b
1
A
0
@
1 1 1 a
0 0 0 b ; a
0 0 0 b ; a
1
A
y p a r a q u e e l s i s t e m a s e a c o m p a t i b l e , b ; a = 0 , y e l c o n j u n t o d e s o l u c i o n e s e s
f ( x y z ) 2 C
3
= x + y + z = a g
s i a 6= b , e l s i s t e m a e s i n c o m p a t i b l e .
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A p l i c a c i o n e s l i n e a l e s 5 1
C a p t u l o 4 A p l i c a c i o n e s l i n e a l e s
1 C o n s i d e r e m o s l a s a p l i c a c i o n e s e n t r e R - e s p a c i o s v e c t o r i a l e s s i g u i e n t e s :
a ) f : R
3
; ! R
3
t a l q u e f ( x y z ) = ( x + y x ; y z ; 1 )
b ) g : R
3
; ! R
2
t a l q u e g ( x y z ) = ( x + z y ; z )
c ) h : R
2
; ! R
3
t a l q u e h ( x y ) = ( x + k y + k x + y ) c o n k 2 R
D e t e r m i n a r s i s o n o n o l i n e a l e s .
S o l u c i o n :
R e c o r d e m o s q u e u n a a p l i c a c i o n f : E ; ! F c o n E F , K - e s p a c i o s v e c t o r i a l e s e s
l i n e a l , s i y s o l o s i
1 ) 8 v w 2 E f ( v + w ) = f ( v ) + f ( w )
2 ) 8 v 2 E 8 2 K f ( v ) = f ( v )
c o m p r o b e m o s s i , p a r a c a d a c a s o , s e v e r i c a n l o s a x i o m a s :
a ) 1 ) s e a n v = ( x
1
y
1
z
1
) w = ( x
2
y
2
z
2
) , e n t o n c e s
v + w = ( x
1
+ x
2
y
1
+ y
2
z
1
+ z
2
)
f ( v + w ) = ( ( x
1
+ x
2
) + ( y
1
+ y
2
) ( x
1
+ x
2
) ; ( y
1
+ y
2
) ( z
1
+ z
2
) ; 1
f ( v ) + f ( w ) = ( x
1
+ y
1
x
1
; y
1
z
1
; 1 ) + ( x
2
+ y
2
x
2
; y
2
z
2
; 1 ) =
= ( ( x
1
+ y
1
) + ( x
2
+ y
2
) ( x
1
; y
1
) + ( x
2
; y
2
) ( z
1
; 1 ) + ( z
2
; 1 ) ) =
= ( ( x
1
+ x
2
) + ( y
1
+ y
2
) ( x
1
+ x
2
) ; ( y
1
+ y
2
) ( z
1
+ z
2
) ; 2 ) ) 6=
6= f ( v + w )
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5 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
l u e g o , n o p u e d e s e r l i n e a l , ( y n o h a c e f a l t a p r o b a r 2 )
b ) 1 ) s e a n v = ( x
1
y
1
z
1
) w = ( x
2
y
2
z
2
) , e n t o n c e s
v + w = ( x
1
+ x
2
y
1
+ y
2
z
1
+ z
2
)
g ( v + w ) = ( ( x
1
+ x
2
) + ( z
1
+ z
2
) ( y
1
+ y
2
) ; ( z
1
+ z
2
) )
g ( v ) + g ( w ) = ( x
1
+ z
1
y
1
; z
1
) + ( x
2
+ z
2
y
2
; z
2
) =
= ( ( x
1
+ z
1
) + ( x
2
+ z
2
) ( y
1
; z
1
) + ( y
2
; z
2
) ) =
= ( ( x
1
+ x
2
) + ( z
1
+ z
2
) ( y
1
+ y
2
) ; ( z
1
+ z
2
) ) = g ( v + w )
2 ) S e a v = ( x
1
y
1
z
1
) , e n t o n c e s 8 2 R v = ( x
1
y
1
z
1
)
g ( v ) = g ( x
1
y
1
z
1
) = ( x
1
+ z
1
y
1
; z
1
)
g ( v ) = ( x
1
+ z
1
y
1
; z
1
) = ( ( x
1
+ z
1
) ( y
1
; z
1
) ) =
= ( x
1
+ z
1
y
1
; z
1
) = g ( v )
l u e g o , g e s l i n e a l .
c ) 1 ) S e a n v = ( x
1
y
1
) w = ( x
2
y
2
) , e n t o n c e s v + w = ( x
1
+ x
2
y
1
+ y
2
)
h ( v + w ) = ( ( x
1
+ x
2
) + k ( y
1
+ y
2
) + k ( x
1
+ x
2
) + ( y
1
+ y
2
) )
h ( v ) + h ( w ) = ( x
1
+ k y
1
+ k x
1
+ y
1
) + ( x
2
+ k y
2
+ k x
2
+ y
2
) =
= ( ( x
1
+ k ) + ( x
2
+ k ) ( y
1
+ k ) + ( y
2
+ k ) ( x
1
+ y
1
) + ( x
2
+ y
2
) ) =
= ( ( x
1
+ x
2
) + 2 k ( y
1
+ y
2
) + 2 k ( x
1
+ x
2
) + ( y
1
+ y
2
) )
p a r a q u e h ( v + w ) = h ( v ) + h ( w ) , e s n e c e s a r i o y s u c i e n t e q u e 2 k = k , e s d e c i r
k = 0
2 ) S e a p u e s k = 0 y s e a v = ( x y ) , e n t o n c e s 8 2 R v = ( x y )
h ( v ) = ( x y x + y )
h ( v ) = ( x y x + y ) = ( x y ( x + y ) ) = ( x y x + y ) = h ( v )
l u e g o , h e s l i n e a l s i y s o l o s i k = 0
N o t a : n o h a c e f a l t a p r o b a r 2 p a r a k 6= 0 y a q u e d e t o d o s m o d o s l a a p l i c a c i o n n o s e r a
l i n e a l .
O b s e r v a c i o n : S e a n E y F d o s e s p a c i o s v e c t o r i a l e s s o b r e K d e d i m e n s i o n e s ( n i t a s ) n
y m r e s p e c . y u n a a p l i c a c i o n f : E = ) F , f ( v ) = f ( x
1
: : : x
n
) = w = ( y
1
: : : y
m
) ,
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A p l i c a c i o n e s l i n e a l e s 5 3
c o n x
i
e y
i
l a s c o o r d e n a d a s d e l o s v e c t o r e s v 2 E y w 2 F r e s p e c t o a b a s e s d e E y
F p r e v i a m e n t e e s c o g i d a s . f e s l i n e a l s i y s o l o s i l a s c o o r d e n a d a s y
i
s o n p o l i n o m i o s
h o m o g e n e o s d e g r a d o 1 e n l a s v a r i a b l e s x
1
: : : x
n
: y
1
= a
1
1
x
1
+ : : : a
1
n
x
n
, , y
m
=
a
m
1
x
1
+ : : : a
m
n
x
n
, y l a m a t r r i z d e l a a p l i c a c i o n e n l a s b a s e s d a d a s e s
A =
0
B
@
a
1
1
: : : a
1
n
a
m
1
: : : a
m
n
1
C
A
e s t o e s c a d a l a d e l a m a t r i z e s t a f o r m a d a p o r l o s c o e c i e n t e s d e l p o l i n o m i o c o r r e s p o n -
d i e n t e .
2 S e a f : R
3
; ! R l i n e a l , d e l a c u a l s e s a b e q u e
f ( 1 2 3 ) = ( 6 4 3 1 )
f ( 2 0 1 ) = ( 3 6 1 2 )
f ( 0 1 0 ) = ( 0 1 2 )
9
>
=
>
H a l l a r l a m a t r i z d e f e n l a b a s e n a t u r a l .
S o l u c i o n :
S i f e
1
e
2
e
3
g e s l a b a s e n a t u r a l , p a r a d a r l a m a t r i z d e f e n d i c h a b a s e n e c e s i t a m o s
c o n o c e r f ( e
1
) f ( e
2
) f ( e
3
) e x p r e s a d o s e n l a b a s e n a t u r a l
( 1 2 3 ) = e
1
+ 2 e
2
+ 3 e
3
f ( 1 2 3 ) = f ( e
1
+ 2 e
2
+ 3 e
3
) = f ( e
1
) + 2 f ( e
2
) + 3 f ( e
3
) =
= ( 6 4 3 1 ) = 6 e
1
+ 4 e
2
+ 3 1 e
3
A n a l o g a m e n t e s e t i e n e q u e
f ( 2 0 1 ) = 2 f ( e
1
) + f ( e
3
) = 3 e
1
+ 6 e
2
+ 1 2 e
3
f ( 0 1 0 ) = f ( e
2
) = e
2
+ 2 e
3
e s d e c i r , t e n e m o s e l s i s t e m a
f ( e
1
) + 2 f ( e
2
) + 3 f ( e
3
) = 6 e
1
+ 4 e
2
+ 3 1 e
3
2 f ( e
1
) + f ( e
3
) = 3 e
1
+ 6 e
2
+ 1 2 e
3
f ( e
2
) = e
2
+ e
3
9
>
=
>
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5 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
q u e , d e s p e j a n d o f ( e
1
) f ( e
2
) f ( e
3
) , t e n e m o s
f ( e
1
) =
3
5
e
1
+
1 6
5
e
2
+
9
5
e
3
= (
3
5
1 6
5
9
5
)
f ( e
2
) = e
2
+ 2 e
3
= ( 0 1 2 )
f ( e
3
) =
9
5
e
1
;
2
5
e
2
+
4 2
5
e
3
= (
9
5
;
2
5
4 2
5
)
l u e g o , l a m a t r i z s e r a
A =
0
B
B
B
@
3
5
0
9
5
1 6
5
1 ;
2
5
9
5
2
4 2
5
1
C
C
C
A
O t r a f o r m a d e r e s o l v e r e l p r o b l e m a :
P u e s t o q u e ( 1 2 3 ) ( 2 0 1 ) ( 0 1 0 ) s o n i n d e p e n d i e n t e s f o r m a n u n a b a s e f v
1
v
2
v
3
g d e
R
3
F i j a n d o e s t a b a s e e n e l e s p a c i o d e p a r t i d a , y j a n d o l a n a t u r a l e n e l e s p a c i o d e l l e g a d a ,
l a m a t r i z d e l a a p l i c a c i o n e n e s t a s b a s e s e s
B =
0
@
6 3 0
4 6 1
3 1 1 2 2
1
A
L a m a t r i z d e p a s o d e l a b a s e f e
1
e
2
e
3
g a l a b a s e f v
1
v
2
v
3
g e s
S
; 1
=
0
@
1 2 0
2 0 1
3 1 0
1
A
; 1
=
1
5
0
@
; 1 0 2
3 0 ; 1
2 5 ; 4
1
A
y t e n e m o s e l d i a g r a m a :
R
3
v
B
; ; ; ; ! R
3
e
S
1
x
?
?
R
3
e
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A p l i c a c i o n e s l i n e a l e s 5 5
L u e g o A = B S
; 1
=
0
B
B
B
@
3
5
0
9
5
1 6
5
1 ;
2
5
9
5
2
4 2
5
1
C
C
C
A
3 S e a E u n e s p a c i o v e c t o r i a l s o b r e K d e d i m e n s i o n n > 1 > Q u e d e s i g u a l d a d v e r i -
c a n l o s r a n g o s d e d o s e n d o m o r s m o s f g d e E t a l e s q u e g f = 0 ?
S o l u c i o n :
S i g f = 0 , e n t o n c e s p a r a t o d o x 2 E s e t i e n e g ( f ( x ) ) = 0 , l o q u e i m p l i c a q u e
I m f e s t a c o n t e n i d o e n K e r g e n t o n c e s d i m I m f d i m K e r g , l o q u e e q u i v a l e a
d e c i r
r a n g o f + r a n g o g n
( y a q u e d i m k e r g + r a n g o g = n )
4 D e m o s t r a r q u e l a a p l i c a c i o n f : R
3
; ! R
2
d e n i d a p o r f ( x y z ) = ( x ; 2 y z +
y ) e s l i n e a l . H a l l a r s u m a t r i z e n l a s b a s e s n a t u r a l e s y s u r a n g o . E n c o n t r a r u n a b a s e d e
K e r g y o t r a d e I m f
S o l u c i o n :
V e a m o s l a l i n e a l i d a d :
S e a n v = ( x
1
y
1
z
1
) w = ( x
2
y
2
z
2
)
f ( v + w ) = f ( x
1
+ x
2
y
1
+ y
2
z
1
+ z
2
) =
= ( ( x
1
+ x
2
) ; 2 ( y
1
+ y
2
) ( z
1
+ z
2
) + ( y
1
+ y
2
) ) =
= ( ( x
1
; 2 y
1
) + ( x
2
; 2 y
2
) ( z
1
+ y
1
) + ( z
2
+ y
2
) ) =
= ( x
1
; 2 y
1
z
1
+ y
1
) + ( x
2
; 2 y
2
z
2
+ y
2
) = f ( v ) + f ( w )
8 2 R s e a v = ( x y z )
f ( v ) = f ( x y z ) = ( x ; 2 y z + y ) =
= ( ( x ; 2 y ) ( z + y ) ) = ( x ; 2 y 2 + y ) = f ( v )
l u e g o , e n e f e c t o e s l i n e a l .
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5 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
H a l l e m o s l a m a t r i z d e l a a p l i c a c i o n , c a l c u l a n d o l a i m a g e n d e l o s v e c t o r e s d e l a b a s e
n a t u r a l d e R
3
f ( 1 0 0 ) = ( 1 0 )
f ( 0 1 0 ) = ( ; 2 1 )
f ( 0 0 1 ) = ( 0 1 )
p o r l o t a n t o
A =
1 ; 2 0
0 1 1
r a n g o A = 2 = d i m I m f = R
2
( = n u m e r o d e v e c t o r e s c o l u m n a d e A q u e s o n
l i n e a l m e n t e i n d e p e n d i e n t e s ) .
d i m K e r f = 3 ; r a n g o f = 3 ; 2 = 1
u n a b a s e d e I m f p u e d e s e r f ( 1 0 ) ( 0 1 ) g
d e t e r m i n e m o s a h o r a u n a d e K e r f
v 2 K e r f s i y s o l o s i f ( v ) = 0 . S e a p u e s ( x y z ) 2 K e r f e n t o n c e s
1 ; 2 0
0 1 1
0
@
x
y
z
1
A
=
x ; 2 y
y + z
=
0
0
P o r l o t a n t o K e r f = f ( x y z ) = x ; 2 y = 0 y + z = 0 g y u n a b a s e p u e d e s e r : t o m a n d o
y = 1 l a s c o m p o n e n t e s x z s o n : x = 2 z = ; 1 y e l v e c t o r d e l a b a s e e s ( 2 1 ; 1 )
D e h e c h o , l o s d o s p r i m e r o s a p a r t a d o s y s e g u n l a o b s e r v a c i o n d e l p r o b l e m a 1 p o d e m o s
r e s o l v e r l o d e l a m a n e r a : f e s l i n e a l y a q u e l a i m a g e n d e u n v e c t o r d a d o e n c o o r d e n a d a s ,
e s u n v e c t o r c u y a s c o o r d e n a d a s s o n p o l i n o m i o s h o m o g e n e o s d e g r a d o 1 e n l a s v a r i a b l e s
d e l v e c t o r i n i c i a l . L a m a t r i z d e l a a p l i c a c i o n v i e n e d a d a p o r l o s c o e c i e n t e s d e d i c h o s
p o l i n o m i o s .
5 U n e n d o m o r s m o f 2 L ( E
2
) t i e n e p o r m a t r i z e n l a b a s e f e
1
e
2
g d e E
2
a
A =
2 ; 3
; 3 2
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A p l i c a c i o n e s l i n e a l e s 5 7
H a l l a r s u m a t r i z e n l a b a s e f e
0
1
e
0
2
g d a d a p o r
2 e
0
1
= e
1
+ e
2
2 e
0
2
= e
2
; e
1
S o l u c i o n :
T e n e m o s e l d i a g r a m a
R
2
B
; ; ; ; ! R
2
'
B
?
?
y
'
1
B
x
?
?
E
2
f
; ; ; ; ! E
2
'
A
x
?
?
'
1
A
?
?
y
R
2
A
; ; ; ; ! R
2
L a r e l a c i o n e n t r e a m b a s m a t r i c e s e s
B = S
; 1
A S
s i e n d o S = '
; 1
A
'
B
l a m a t r i z c a m b i o d e b a s e q u e p a r a l a b a s e f e
1
e
2
g
S =
0
@
1
2
;
1
2
1
2
1
2
1
A
N e c e s i t a m o s c o n o c e r S
; 1
S
; 1
=
1 1
; 1 1
y p o r l o t a n t o
B =
1 1
; 1 1
2 ; 3
; 3 2
1
2
;
1
2
1
2
1
2
=
; 1 0
0 5
6 S e a f : R
2
; ! R
3
u n a a p l i c a c i o n l i n e a l d e n i d a p o r
f ( x y ) = ( 2 x ; y x + y 2 y ; x )
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5 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
a ) D a r l a m a t r i z d e f e n l a s b a s e s n a t u r a l e s d e R
2
y R
3
r e s p e c t i v a m e n t e c a l c u l a r
f ( 3
1
2
)
b ) D a r u n a b a s e , y l a d i m e n s i o n d e K e r f y d e I m f
c ) D a r l a m a t r i z d e f e n l a s b a s e s f v
1
v
2
g , f u
1
u
2
u
3
g , s i e n d o
v
1
= ( 2 1 )
v
2
= ( 0 3 )
u
1
= ( 1 1 1 )
u
2
= ( 2 0 1 )
u
3
= ( 0 0 2 )
9
>
=
>
C a l c u l a r f (
1
2
v
1
+ ( ;
1
3
) v
2
)
S o l u c i o n :
a ) f ( 1 0 ) = ( 2 1 ; 1 ) f ( 0 1 ) = ( ; 1 1 2 ) , l u e g o l a m a t r i z d e f e n l a s b a s e s n a t u -
r a l e s e s
A =
0
@
2 ; 1
1 1
; 1 2
1
A
y
f ( 3
1
2
) =
0
@
2 ; 1
1 1
; 1 2
1
A
3
1
2
=
0
@
1 1
2
7
2
; 2
1
A
b ) K e r f = f ( x y ) 2 R
2
= f ( x y ) = 0 g , l u e g o
0
@
2 ; 1
1 1
; 1 2
1
A
x
y
= ( 0 0 0 ) )
2 x ; y = 0
x + y = 0
; x + 2 y = 0
9
>
=
>
S i s t e m a c o m p a t i b l e y d e t e r m i n a d o l u e g o K e r f = f ( 0 0 ) g y d i m K e r f = 0 , p o r l o
t a n t o
d i m I m f = d i m R
2
; d i m K e r f = 2 y
I m f = ( 2 1 ; 1 ) ( ; 1 1 2 )
c ) T e n e m o s e l d i a g r a m a :
R
2
A
; ; ; ; ! R
3
S
x
?
?
T
1
?
?
y
R
2
f v g
B
; ; ; ; ! R
3
f u g
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A p l i c a c i o n e s l i n e a l e s 5 9
s i e n d o S =
2 0
1 3
l a m a t r i z d e p a s o d e l a b a s e f v
1
v
2
g a l a n a t u r a l y T =
0
@
1 2 0
1 0 0
1 1 2
1
A
l a m a t r i z d e p a s o d e l a b a s e f u
1
u
2
u
3
g a l a n a t u r a l . N e c e s i t a m o s
T
; 1
T
; 1
=
1
4
0
@
0 4 0
2 ; 2 0
; 1 ; 1 2
1
A
y p o r l o t a n t o
B = T
; 1
A S =
0
@
3 3
0 ; 3
;
3
2
3
1
A
n a l m e n t e , s i h a c e m o s
f (
3
2
v
1
+ ( ;
1
3
v
2
) ) =
0
@
3 3
0 ; 3
;
3
2
3
1
A
3
2
;
1
3
=
0
@
7
2
1
;
1 3
4
1
A
=
=
7
2
u
1
+ u
2
;
1 3
4
u
3
o b s e r v a m o s q u e
3
2
v
1
+ ( ;
1
3
) v
2
=
3
2
( 2 1 ) ;
1
3
( 0 3 ) = ( 3
1
2
) l u e g o
7
2
u
1
+ u
2
;
1 3
4
u
3
= (
1 1
2
7
2
; 2 )
7 S e a n E F G t r e s K - e s p a c i o s v e c t o r i a l e s ( d e d i m e n s i o n n i t a ) , y s e a n
f : E ; ! G
g : F ; ! G
d o s a p l i c a c i o n e s l i n e a l e s . D e m o s t r a r q u e e s c o n d i c i o n n e c e s a r i a y s u c i e n t e p a r a q u e
e x i s t a a l m e n o s u n a a p l i c a c i o n h : E ; ! F t a l q u e g h = f , q u e I m f I m g
S o l u c i o n :
C o n s i d e r e m o s e l d i a g r a m a
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6 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
E
h
; ; ; ; ! F
f
?
?
y
g
G
V e a m o s q u e l a c o n d i c i o n e s n e c e s a r i a . S e a y 2 I m f e x i s t e p u e s x 2 E , t a l q u e
f ( x ) = y
s i g h = f s e t i e n e g ( h ( x ) ) = f ( x ) = y , e s d e c i r , e x i s t e z 2 F ( z = h ( x ) ) , t a l
q u e g ( z ) = y . P o r l o t a n t o , I m f I m g
V e a m o s q u e l a c o n d i c i o n e s s u c i e n t e . C o n s i d e r e m o s K e r g F y s e a F
1
F , t a l
q u e F = K e r g F
1
D a d o x 2 E , v a m o s a d e n i r h ( x ) . S e a f ( x ) 2 I m f I m g , l u e g o e x i s t e y 2 F t a l
q u e g ( y ) = f ( x ) c o n y = y
1
+ y
2
, y
1
2 K e r g , y
2
2 F
1
. L u e g o g ( y ) = g ( y
2
)
D e n i m o s h ( x ) = y
2
h e s t a b i e n d e n i d o , p u e s s e a n y = y
1
+ y
2
y = y
1
+ y
2
t a l e s
q u e g ( y ) = g ( y ) e n t o n c e s y ; y 2 K e r g p u e s t o q u e y
2
; y
2
2 F
1
y y ; y 2 K e r g ,
s e t i e n e q u e y
2
; y
2
2 K e r g \ F
1
= f 0 g l u e g o y
2
= y
2
8 S e a R x ] e l e s p a c i o d e l o s p o l i n o m i o s a c o e c i e n t e s r e a l e s . D e n i m o s D M :
R x ; ! R x ] m e d i a n t e
1 ) D ( P ( x ) ) = P
0
( x ) p o l i n o m i o d e r i v a d o .
2 ) M ( P ( x ) ) = x P ( x )
a ) P r o b a r q u e D y M s o n l i n e a l e s .
b ) > E s D n i l p o t e n t e ? ( d i r e m o s q u e f 2 L ( E ) e s n i l p o t e n t e s i e x i s t e n 2 N , t a l q u e
f
n
= 0 ) .
c ) P r o b a r q u e D M ; M D = I
d ) D e d u c i r d e e l l o q u e ( D M )
2
= D
2
M
2
; D M
S o l u c i o n :
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A p l i c a c i o n e s l i n e a l e s 6 1
a ) V e a m o s q u e s e c u m p l e n l a s d o s c o n d i c i o n e s
D ( p ( x ) + q ( x ) ) = ( p ( x ) + q ( x ) )
0
= p
0
( x ) + q
0
( x ) = D ( p ( x ) + D ( q ( x ) )
D ( p ( x ) ) = ( p ( x ) )
0
= p
0
( x ) = D ( p ( x ) )
l u e g o D e s l i n e a l .
M ( p ( x ) + q ( x ) ) = x ( p ( x ) + q ( x ) ) = x p ( x ) + x q ( x ) = M ( p ( x ) ) + M ( q ( x ) )
M ( p ( x ) ) = x ( p ( x ) ) = x p ( x ) = M ( p ( x ) )
l u e g o M e s l i n e a l .
b ) S i e x i s t e n 2 N t a l q u e D
n
= 0 , i m p l i c a q u e p a r a t o d o p ( x ) 2 R x ] , e s
D
n
p ( x ) = 0
S e a q ( x ) = a
n + 1
x
n + 1
+ + a
0
, c o n a
n + 1
6= 0 ,
D
n
( q ( x ) ) = D
n ; 1
( D ( q ( x ) ) = D
n ; 1
( a
n + 1
( n + 1 ) x
n
+ + a
1
) =
= = a
n + 1
( n + 1 ) ! 6= 0
c o n t r a d i c c i o n . L u e g o D n o e s n i l p o t e n t e .
N o t e s e q u e R x ] e s d e d i m e n s i o n i n n i t a , y q u e s i n o s r e s t r i n g i m o s a D
R
n
x
:
R
n
x ; ! R
n
x ] e n t o n c e s s q u e e s n i l p o t e n t e , p u e s D
n + 1
= 0
c ) D a d a s f g 2 E n d ( E ) e n t o n c e s f = g , 8 x 2 E f ( x ) = g ( x ) , e n n u e s t r o c a s o :
( D M ; M D ) ( p ( x ) ) = D ( M ( p ( x ) ) ) ; M ( D ( p ( x ) ) ) =
D ( x p ( x ) ) ; M ( p
0
( x ) ) = p ( x ) + x p
0
( x ) ; x p
0
( x ) = p ( x ) = I ( p ( x ) )
l u e g o D M ; M D = I
d ) ( D M )
2
= ( D M ) ( D M ) = D ( M D ) M = D ( D M ; I ) M = ( D
2
M ; D ) M = D
2
M
2
;
D M
9 S e a E e l e s p a c i o d e l a s m a t r i c e s c u a d r a d a s a c o e c i e n t e s e n C d e o r d e n 2 . D e n -
i m o s u n a a p l i c a c i o n f d e E e n C d e l a f o r m a
f (
a b
c d
) ) = a + d
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6 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
a ) P r o b a r q u e f e s l i n e a l .
b ) P r o b a r q u e s i A B s o n d o s e l e m e n t o s c u a l e s q u i e r a d e E , s e t i e n e f ( A B ) =
f ( B A )
c ) D e m o s t r a r q u e e s i m p o s i b l e e n c o n t r a r d o s e l e m e n t o s d e E t a l e s q u e
A B ; B A = I
d ) D a r u n a b a s e d e K e r f
S o l u c i o n :
a ) S e a n A =
a b
c d
A
0
=
a
0
b
0
c
0
d
0
f ( A + A
0
) = f (
a + a
0
b + b
0
c + c
0
d + d
0
) = a + a
0
+ b + b
0
=
= a + d + a
0
+ d
0
= f ( A ) + f ( A
0
)
f ( A ) = f (
a b
c d
) = a + d = ( a + d ) = f ( A )
l u e g o f e s l i n e a l .
O t r a f o r m a : e s c o g i d a s b a s e s e
1
=
1 0
0 0
, e
2
=
0 1
0 0
, e
3
=
0 0
1 0
, e
4
=
0 0
0 1
, p a r a
E y 1 p a r a C , l a a p l i c a c i o n s e e x p r e s a :
f ( a b c d ) = a + d
y a h o r a p o d e m o s a p l i c a r l a o b s e r v a c i o n d a d a e n e l p r o b l e m a 1 .
b ) S e a n A =
a b
c d
B =
a
0
b
0
c
0
d
0
, e n t o n c e s
A B =
a a
0
+ b c
0
a b
0
+ b d
0
c a
0
+ d c
0
c b
0
+ d d
0
B A =
a
0
a + b
0
c a
0
b + b
0
d
c
0
a + d
0
c c
0
b + d
0
d
f ( A B ) = a a
0
+ b c
0
+ c b
0
+ d d
0
= a
0
a + b
0
c + c
0
b + d
0
d = f ( B A )
c ) P u e s t o q u e f ( A B ) = f ( B A ) s e t i e n e q u e f ( A B ; B A ) = 0 , o s e a q u e , p a r a
t o d o A B 2 E s e t i e n e q u e A B ; B A 2 k e r f S i I = A B ; B A p a r a u n a s c i e r t a s
m a t r i c e s A B s e t e n d r a I 2 k e r f , p e r o f ( I ) = 2 6= 0 l u e g o , p a r a t o d o A B 2 E
e s A B ; B A 6= I
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A p l i c a c i o n e s l i n e a l e s 6 3
( C o m p a r a r d i c h o r e s u l t a d o c o n e l h a l l a d o e n e l a p a r t a d o c , d e l p r o b l e m a a n t e r i o r ) .
d ) S i A =
a b
c d
2 k e r f e s a = ; d l u e g o
A =
a b
c d
= a
1 0
0 ; 1
+ b
0 1
0 0
+ c
0 0
1 0
= a A
1
+ b A
2
+ c A
3
A
1
A
2
A
3
2 k e r f , g e n e r a n k e r f y s o n l i n e a l m e n t e i n d e p e n d i e n t e s , l u e g o s o n b a s e
d e k e r f
1 0 S e a E u n e s p a c i o v e c t o r i a l s o b r e K , c u e r p o c o n m u t a t i v o d e c a r a c t e r s t i c a d i s -
t i n t a d e d o s . D i r e m o s q u e p 2 E n d ( E ) e s u n p r o y e c t o r s i y s o l o s i p
2
= p
a ) C o m p r o b a r q u e : p e s p r o y e c t o r , I ; p e s p r o y e c t o r .
b ) C o m p r o b a r q u e : p e s p r o y e c t o r ) I m p K e r p = E
c ) > E s c i e r t o e l r e c p r o c o d e b ?
d ) S i p
1
p
2
s o n p r o y e c t o r e s , c o m p r o b a r q u e p
1
+ p
2
e s p r o y e c t o r s i y s o l o s i p
1
p
2
=
p
2
p
1
= 0
S o l u c i o n :
a ) ) ) ( I ; p )
2
= I
2
+ p
2
; I p ; p I = I
2
+ p
2
; 2 p =
( a )
I + p ; 2 p = I ; p , l u e g o s i
p e s p r o y e c t o r I ; p t a m b i e n l o e s .
( a ) p o r s e r p p r o y e c t o r
( ) P u e s t o q u e ( I ; p )
2
= I ; p , s e t i e n e q u e I + p
2
; 2 p = I ; p , l u e g o p
2
; p = 0 ,
e s d e c i r , p
2
= p
b ) S e a x 2 E . C o n s i d e r e m o s x ; p ( x ) s e t i e n e q u e
p ( x ; p ( x ) ) = p ( x ) ; p
2
( x ) = p ( x ) ; p ( x ) = 0
l u e g o x ; p ( x ) 2 K e r f
O b v i a m e n t e , x = p ( x ) + x ; p ( x ) 2 I m p + K e r p
) E I m p + K e r p E ) I m p + K e r p = E
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6 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
V e a m o s q u e l a s u m a e s d i r e c t a : s e a x 2 K e r p \ I m p s e t i e n e p ( x ) = 0 y e x i s t e
y 2 E , t a l q u e p ( y ) = x , l u e g o 0 = p ( x ) = p
2
( y ) =
( a )
p ( y ) = x
E n t o n c e s E = I m p K e r p
( a ) p o r s e r p p r o y e c t o r
c ) C o n s i d e r e m o s E = R
2
y f t a l q u e s u m a t r i z e n l a b a s e n a t u r a l s e a
2 0
0 0
S e t i e n e I m f = ( 1 0 ) K e r f = ( 0 1 ) ] , l u e g o R
2
= I m f K e r f . S i n e m b a r g o ,
2 0
0 0
2
=
4 0
0 0
6=
2 0
0 0
, l u e g o f n o e s p r o y e c t o r .
N o t a : E n u n e s p a c i o d e d i m e n s i o n n i t a E s e t i e n e s i e m p r e q u e
d i m E = d i m I m f + d i m K e r f
p a r a t o d o f 2 E n d E , p e r o e s t o n o i m p l i c a E = I m f K e r f . P a r a e l l o v e a m o s u n
e j e m p l o :
S e a R
2
y f t a l q u e e n l a b a s e n a t u r a l s e a
0 1
0 0
T e n e m o s I m f = ( 0 1 ) ] y K e r f = ( 0 1 ) ] , l u e g o I m f = K e r f y n o p u e d e n f o r m a r
s u m a r d i r e c t a p e r o
d i m I m f + d i m K e r f = 1 + 1 = 2 = d i m R
2
d ) S e a n p
1
p
2
p r o y e c t o r e s . S i p
1
+ p
2
e s p r o y e c t o r ) ( p
1
+ p
2
)
2
= p
1
+ p
2
p e r o
( p
1
+ p
2
)
2
= p
2
1
+ p
2
2
+ p
1
p
2
+ p
2
p
1
= p
1
+ p
2
+ p
1
p
2
+ p
2
p
1
l u e g o p
1
p
2
+ p
2
p
1
= 0 l u e g o p
1
p
2
= ; p
2
p
1
, p o r l o t a n t o
p
1
( p
1
p
2
) = ; p
1
( p
2
p
1
)
p
1
p
2
= p
2
1
p
2
= ; ( p
1
p
2
) p
1
y c o m p o n i e n d o e s t a u l t i m a i g u a l d a d p o r p
1
p o r l a d e r e c h a t e n e m o s
( p
1
p
2
) p
1
= ; ( ( p
1
p
2
) p
1
) p
1
= ; ( p
1
p
2
) ( p
1
p
1
) = ; ( p
1
p
2
) p
1
l u e g o p
1
p
2
p
1
= ; p
1
p
2
p
1
) 2 p
1
p
2
p
1
= 0 )
( a )
p
1
p
2
p
1
= 0 p o r l o q u e
p
1
p
2
= ; p
1
p
2
p
1
= 0
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A p l i c a c i o n e s l i n e a l e s 6 5
l u e g o p
1
p
2
= p
2
p
1
= 0 , y r e c p r o c a m e n t e , s i p
1
p
2
s o n p r o y e c t o r e s y p
1
p
2
=
p
2
p
1
= 0 , s e t i e n e
( p
1
+ p
2
)
2
= p
2
1
+ p
2
2
+ p
1
p
2
+ p
2
p
1
= p
1
+ p
2
l u e g o p
1
+ p
2
, e s u n p r o y e c t o r .
( a ) a q u e s d o n d e i n t e r v i e n e l a h i p o t e s i s d e c a r a c K 6= 2
1 1 S e a n u
1
= ( 2 ; 1 1 ) u
2
= ( 1 1 0 ) u
3
= ( 1 ; 2 3 ) u
4
= ( 6 ; 1 6 ) v e c t o r e s
d e R
3
y s e a f : R
3
; ! R
2
u n a a p l i c a c i o n l i n e a l d e l a q u e c o n o c e m o s f ( u
1
) =
( 1 ; 1 ) f ( u
2
) = ( 4 1 ) y f ( u
3
) = ( 3 1 )
a ) > E s p o s i b l e d e t e r m i n a r f ( u
4
) ? , > P o r q u e ?
b ) L a a p l i c a c i o n f > s e r a i n y e c t i v a ? > s e r a e x h a u s t i v a ?
c ) C a l c u l a r l a m a t r i z d e f e n l a s b a s e s n a t u r a l e s d e R
3
y R
2
r e s p e c t i v a m e n t e .
d ) D e t e r m i n a r u n a b a s e d e K e r f
S o l u c i o n :
a ) E s p o s i b l e h a l l a r f ( u
4
) y a q u e f u
1
u
2
u
3
g f o r m a n b a s e d e R
3
( E n e f e c t o ,
2 1 1
; 1 1 ; 2
1 0 3
= 6 6= 0 ) , p o r l o t a n t o
u
4
= a u
1
+ b u
2
+ c u
3
y f ( u
4
) = a f ( u
1
) + b f ( u
2
) + c f ( u
3
)
b ) f n o p u e d e s e r i n y e c t i v a , y a q u e
d i m K e r f = d i m R
3
; d i m I m f 3 ; 2 = 1 6= 0
( d i m I m f 2 p u e s t o q u e I m f R
2
)
f s e r a e x h a u s t i v a e n c a s o d e q u e d i m I m f = 2 . V e a m o s s i e s a s : l a m a t r i z d e f e n
l a b a s e f u
1
u
2
u
3
g d e R
3
y l a n a t u r a l d e R
2
e s
A =
1 4 3
; 1 1 1
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6 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
y r a n g o A = 2 = r a n g o
1 4 3
; 1 1 1
= r a n g o
1 0 0
; 1 5 4
=
= r a n g o
1 0 0
; 1 1 0
c ) L a m a t r i z d e f e n l a s b a s e s n a t u r a l e s s e r a
R
3
u
A
; ; ; ; ! R
2
S
?
?
y
A
0
R
3
e
S e s l a m a t r i z d e c a m b i o d e b a s e d e f u
1
u
2
u
3
g a l a n a t u r a l f e
1
e
2
e
3
g , y A
0
=
A S
; 1
c o n
S =
0
B
B
B
@
2 1 1
; 1 1 ; 2
1 0 3
1
C
C
C
A
y S
; 1
=
0
B
B
B
@
1
2
;
1
2
;
1
2
1
6
5
6
1
2
;
1
6
1
6
1
2
1
C
C
C
A
A
0
=
1 4 3
; 1 1 1
0
B
B
B
@
1
2
;
1
2
;
1
2
1
6
5
6
1
2
;
1
6
1
6
1
2
1
C
C
C
A
=
0
@
2
3
2 0
6
3
;
1
2
3
2
3
2
1
A
d ) K e r f = f ( x y z ) 2 R
3
= f ( x y z ) = 0 g
0
@
2
3
2 0
6
3
;
1
2
3
2
3
2
1
A
0
@
x
y
z
1
A
=
0
0
4 x + 2 0 y + 1 8 z = 0
; x + 3 y + 3 z = 0
S i s t e m a c o m p a t i b l e i n d e t e r m i n a d o d e r a n g o = 2 c u y o c o n j u n t o d e s o l u c i o n e s e s
K e r f = f ( x y z ) 2 R
3
= y = ;
1 5
1 6
z x =
3
1 6
z g = ( 3 1 5 1 6 ) ]
1 2 E n c o n t r a r l o s v a l o r e s d e a p a r a l o s c u a l e s e l e n d o m o r s m o d e R
3
d a d o p o r
f ( x y z ) = ( x + a y ; a z a x + y + z ; a x + a y + z ) e s u n a u t o m o r s m o .
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A p l i c a c i o n e s l i n e a l e s 6 7
S o l u c i o n :
f s e r a a u t o m o r s m o s i e l d e t e r m i n a n t e d e s u m a t r i z a s o c i a d a , e n c u a l q u i e r b a s e , e s
d i s t i n t o d e c e r o . B u s q u e m o s p u e s l a m a t r i z d e f e n l a b a s e n a t u r a l ( p o r e j e m p l o ) .
f ( 1 0 0 ) = ( 1 a ; a )
f ( 0 1 0 ) = ( a 1 a )
f ( 0 0 1 ) = ( ; a 1 1 )
l u e g o
A =
0
@
1 a ; a
a 1 1
; a a 1
1
A
y d e t A = 1 ; a ; 3 a
2
; a
3
= ; ( a + 1 ) ( a + 1 ;
p
2 ) ( a + 1 +
p
2 ) , l u e g o d e t A 6= 0 s i
y s o l o s i a e s d i s t i n t o d e
; 1 ; 1 +
p
2 ; 1 ;
p
2
1 3 S e a l a m a t r i z
A =
0
@
0 2 1
0 0 3
0 0 0
1
A
a s o c i a d a a l a a p l i c a c i o n l i n e a l f d e n i d a s o b r e R
3
r e s p e c t o l a b a s e n a t u r a l .
a ) H a l l a r l o s s u b e s p a c i o s I m f y K e r f
b ) H a l l a r u n a b a s e d e R
3
p a r a l a c u a l l a m a t r i z d e f e n d i c h a b a s e s e a
B =
0
@
0 1 0
0 0 1
0 0 0
1
A
S o l u c i o n :
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6 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
a ) R e c o r d a n d o l a d e n i c i o n d e I m f y d e K e r f :
I m f = f y 2 R
3
= 9 x 2 R
3
c o n f ( x ) = y g =
= f ( 1 0 0 ) f ( 0 1 0 ) f ( 0 0 1 ) ] =
= ( 0 0 0 ) ( 2 0 0 ) ( 1 3 0 ) ] = ( 2 0 0 ) ( 1 3 0 )
K e r f = f x 2 R
3
= f ( x ) = 0 g
0
@
0 2 1
0 0 3
0 0 0
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
,
2 y + z = 0
3 z = 0
)
) K e r f = ( 1 0 0 )
D e h e c h o , o b s e r v a n d o l a m a t r i z A q u e e s d e r a n g o 2 y l a p r i m e r a c o l u m n a e s i d e n t i c a m e n t e
n u l a , y a p o d e m o s a r m a r q u e K e r f = ( 1 0 0 )
b ) B u s c a m o s v
1
= ( x
1
x
2
x
3
) v
2
= ( y
1
y
2
y
3
) v
3
= ( z
1
z
2
z
3
) t a l e s q u e f o r m e n
b a s e , y s i
S =
0
@
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
1
A
e n t o n c e s S B = A S
0
@
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
1
A
0
@
0 1 0
0 0 1
0 0 0
1
A
=
0
@
0 2 1
0 0 3
0 0 0
1
A
0
@
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
1
A
p o r l o t a n t o
x
2
= x
3
= y
3
= 0
2 y
2
= x
1
3 z
3
= y
2
2 z
2
+ z
3
= y
1
9
>
>
>
=
>
>
>
y p o d e m o s t o m a r :
v
1
= ( 6 0 0 ) v
2
= ( 1 3 0 ) v
3
= ( 0 0 1 )
1 4 S e c o n s i d e r a l a a p l i c a c i o n f
k
: R
3
; ! R
3
d e n i d a p o r
f
k
( x y z ) = ( ( 2 ; k ) x + ( k ; 1 ) y 2 ( 1 ; k ) x + ( 2 k ; 1 ) y k z )
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A p l i c a c i o n e s l i n e a l e s 6 9
a ) D e t e r m i n a r l a m a t r i z d e f
k
a s o c i a d a a l a b a s e n a t u r a l d e R
3
b ) D e t e r m i n a r K e r f
k
c ) S u p u e s t o k ( k ; 1 ) 6= 0 , d e m o s t r a r q u e l a m a t r i z M
k
p u e d e p o n e r s e d e l a f o r m a
M
k
= A + k B d o n d e A y B s o n d o s m a t r i c e s a d e t e r m i n a r , y d a r l a e x p r e s i o n d e
M
2
k
e n f u n c i o n d e A B y k y d e d u c i r d e e l l o u n a e x p r e s i o n p a r a M
n
k
S o l u c i o n :
a )
f ( 1 0 0 ) = ( 2 ; k 2 ( 1 ; k ) 0 )
f ( 0 1 0 ) = ( k ; 1 2 k ; 1 0 )
f ( 0 0 1 ) = ( 0 0 k )
l u e g o
M
k
=
0
@
2 ; k k ; 1 0
2 ( 1 ; k ) 2 k ; 1 0
0 0 k
1
A
b ) O b s e r v a m o s q u e d e t M
k
= k
2
, l u e g o , s i k 6= 0 , e s r a n g o f
k
= 3 y , p o r t a n t o ,
K e r f
k
= f 0 g
S e a p u e s k = 0
M
0
=
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
y e l K e r f
0
e s :
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
) 2 x ; y = 0
L u e g o K e r f
0
= ( 1 2 0 ) ( 0 0 1 )
c )
M
k
=
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
+ k
0
@
; 1 1 0
; 2 2 0
0 0 1
1
A
= A + k B
M
2
k
= ( A + k B )
2
= A
2
+ k
2
B
2
+ k A B + k B A
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7 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
O b s e r v a m o s q u e
A
2
=
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
=
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
= A
B
2
=
0
@
; 1 1 0
; 2 2 0
0 0 1
1
A
0
@
; 1 1 0
; 2 2 0
0 0 1
1
A
=
0
@
; 1 1 0
; 2 2 0
0 0 1
1
A
= B
A B =
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
0
@
; 1 1 0
; 2 2 0
0 0 1
1
A
=
0
@
0 0 0
0 0 0
0 0 0
1
A
= 0
B A =
0
@
; 1 1 0
; 2 2 0
0 0 1
1
A
0
@
2 ; 1 0
2 ; 1 0
0 0 0
1
A
=
0
@
0 0 0
0 0 0
0 0 0
1
A
= 0
l u e g o , M
2
k
= A + k
2
B , y p o r l o t a n t o , p o r i n d u c c i o n s e c o n c l u y e q u e M
n
k
= A + k
n
B ,
v e a m o s l o :
E s v a l i d o p a r a n = 1 , s u p u e s t o c i e r t o p a r a n ; 1 : M
n ; 1
k
= A + k
n ; 1
B v e a m o s l o
p a r a n :
M
n
k
= M
k
M
n ; 1
k
= M
k
( A + k
n ; 1
B ) = ( A + k B ) ( A + k
n ; 1
B ) =
= A
2
+ k
n ; 1
A B + k B A + k
n
B
2
= A + k
n ; 1
0 + k 0 + k
n
B = A + k
n
B
N o t a : s i k = 0 , e n t o n c e s B p o d r a s e r c u a l q u i e r m a t r i z , y p o r t a n t o n o t e n d r a p o r
q u e s e r A B = B A = 0 . Y s i k = 1 , M
k
= I y M
n
k
= I , s i b i e n l a e x p r e s i o n h a l l a d a
t i e n e s e n t i d o .
1 5 S e a R
n
x ] e l e s p a c i o v e c t o r i a l d e l o s p o l i n o m i o s d e g r a d o m e n o r o i g u a l q u e
n . C o n s i d e r e m o s l o s e n d o m o r s m o s f D : R
n
x ; ! R
n
x ] s i e n d o D e l o p e r a d o r
d e r i v a d a : D ( p ( x ) ) = p
0
( x ) y f t a l q u e f ( p ( x ) ) = p ( x ) ; p
0
( x ) . D e m o s t r a r q u e
e x i s t e f
; 1
, y q u e s e p u e d e p o n e r e n f u n c i o n d e l o p e r a d o r D
S o l u c i o n :
S e a p ( x ) 2 K e r f ) f ( p ( x ) ) = 0 = p ( x ) ; p
0
( x ) , l u e g o p ( x ) = p
0
( x ) . P e r o
g r a d o p
0
( x ) g r a d o p ( x ) , y v a l e l a i g u a l d a d , s i y s o l o s i g r a d o p ( x ) = 0 , l u e g o
p ( x ) = a p o l i n o m i o c o n s t a n t e , p e r o p
0
( x ) = ( a )
0
= 0 . L u e g o a = 0 y p ( x ) = 0 .
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A p l i c a c i o n e s l i n e a l e s 7 1
P o r l o t a n t o , f e s i n y e c t i v a y t o d a a p l i c a c i o n l i n e a l i n y e c t i v a d e u n e s p a c i o v e c t o r i a l d e
d i m e n s i o n n i t a e n s m i s m o e s e x h a u s t i v a , y p o r t a n t o , e x i s t e f
; 1
S i f
; 1
s e p u e d e p o n e r e n f u n c i o n d e D e s t a h a d e s e r :
f
; 1
=
0
D
0
+ +
n
D
n
y a q u e D
n + 1
= 0
y h a d e c u m p l i r s e q u e f
; 1
f = I
N o t a r q u e : f = I ; D
I = f
; 1
f = (
0
D
0
+ +
n
D
n
) ( I ; D ) =
=
0
D
0
+ +
n
D
n
;
0
D
1
; ;
D
n + 1
n
=
=
0
I + (
1
;
0
) D
1
+ (
2
;
1
) D
2
+ (
n
;
n ; 1
) D
n ; 1
p o r l o t a n t o
0
= 1
1
;
0
= 0
n
;
n ; 1
= 0
9
>
>
>
=
>
>
>
)
i
= 1 i = 1 n
y f
; 1
= D
0
+ + D
n
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D e t e r m i n a n t e s 7 3
C a p t u l o 5 D e t e r m i n a n t e s
1 D a d a s l a s p e r m u t a c i o n e s s = ( 4 3 1 2 ) t = ( 1 2 4 3 ) , d e t e r m i n a r l a s p e r m u t a -
c i o n e s s t t s s
; 1
t
; 1
, a s c o m o e l s i g n o d e c a d a u n a d e e l l a s .
S o l u c i o n :
R e c o r d a n d o q u e s = ( a b c d ) s i g n i c a s ( 1 ) = a , s ( 2 ) = b , s ( 3 ) = c , s ( 4 ) = d ,
t e n e m o s
s t ( 1 ) = s ( t ( 1 ) ) = s ( 1 ) = 4
s t ( 2 ) = s ( t ( 2 ) ) = s ( 2 ) = 3
s t ( 3 ) = s ( t ( 3 ) ) = s ( 4 ) = 2
s t ( 4 ) = s ( t ( 4 ) ) = s ( 3 ) = 1
9
>
>
>
=
>
>
>
s t = ( 4 3 2 1 )
t s ( 1 ) = t ( s ( 1 ) ) = t ( 4 ) = 3
t s ( 2 ) = t ( s ( 2 ) ) = t ( 3 ) = 4
t s ( 3 ) = t ( s ( 3 ) ) = t ( 1 ) = 1
t s ( 4 ) = t ( s ( 4 ) ) = t ( 2 ) = 2
9
>
>
>
=
>
>
>
t s = ( 3 4 1 2 )
D e s
; 1
( 1 ) = i t e n e m o s s ( i ) = 1 l u e g o i = 3
D e s
; 1
( 2 ) = i t e n e m o s s ( i ) = 2 l u e g o i = 4
D e s
; 1
( 3 ) = i t e n e m o s s ( i ) = 3 l u e g o i = 2
D e s
; 1
( 4 ) = i t e n e m o s s ( i ) = 4 l u e g o i = 1
9
>
>
>
=
>
>
>
s
; 1
= ( 3 4 2 1 )
D e t
; 1
( 1 ) = i t e n e m o s t ( i ) = 1 l u e g o i = 1
D e t
; 1
( 2 ) = i t e n e m o s t ( i ) = 2 l u e g o i = 2
D e t
; 1
( 3 ) = i t e n e m o s t ( i ) = 3 l u e g o i = 4
D e t
; 1
( 4 ) = i t e n e m o s t ( i ) = 4 l u e g o i = 3
9
>
>
>
=
>
>
>
t
; 1
= ( 1 2 4 3 )
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7 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
V e a m o s c u a l e s e l s i g n o d e c a d a u n a d e e s t a s p e r m u t a c i o n e s
N ( s ) = 3 + 2 = 5 l u e g o s e s i m p a r : " ( s ) = ; 1
N ( t ) = 1 l u e g o t e s i m p a r : " ( t ) = ; 1
" ( s t ) = " ( s ) " ( t ) = ( ; 1 ) ( ; 1 ) = 1
" ( t s ) = " ( t ) " ( s ) = ( ; 1 ) ( ; 1 ) = 1
" ( s
; 1
) = " ( s ) = ; 1
" ( t
; 1
) = " ( t ) = ; 1
2 H a l l a r e l v a l o r d e l d e t e r m i n a n t e
A =
3 1 2
0 1 1
1 0 1
S o l u c i o n :
R e c o r d a n d o l a d e n i c i o n d e d e t e r m i n a n t e :
s i A = ( a
i
j
) d e t A =
X
s
" ( s ) a
s
1
1
a
s
2
2
a
s
3
3
s e t i e n e
A = + 3 1 1 + 0 0 2 + 1 1 1 ; 1 1 2 ; 3 0 1 ; 0 1 1 = 3 + 1 ; 2 = 2
3 H a l l a r e l v a l o r d e l d e t e r m i n a n t e
A =
3 ; 2 4 5
2 0 5 2
1 3 ; 1 2
2 5 2 ; 3
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D e t e r m i n a n t e s 7 5
a ) P o r l a r e g l a d e L a p l a c e , p o r e j e m p l o , p o r l o s m e n o r e s d e l a s d o s p r i m e r a s
c o l u m n a s .
b ) P o r l o s e l e m e n t o s d e u n a l n e a , p o r e j e m p l o , d e l a p r i m e r a l a .
c ) O b t e n i e n d o \ a p r i o r i " c e r o s e n u n a l n e a y d e s a r r o l l a n d o l u e g o p o r l o s e l e m e n t o s d e
e s t a ( r e d u c c i o n d e l o r d e n ) .
S o l u c i o n :
a )
A =
3 ; 2 4 5
2 0 5 2
1 3 ; 1 2
2 5 2 ; 3
( 1 )
( 2 )
( 3 )
( 4 )
H a y q u e f o r m a r s u m a s d e p r o d u c t o s d e d e t e r m i n a n t e s 2 2 , e x t r a d o s d e A
d e m a n e r a q u e l a s d o s c o l u m n a s d e l p r i m e r f a c t o r s e c o r r e s p o n d a n c o n l a p r i m e r a y
s e g u n d a c o l u m n a s d e A y l a s d o s c o l u m n a s d e l s e g u n d o f a c t o r s e c o r r e s p o n d a n c o n
l a t e r c e r a y c u a r t a c o l u m n a s d e A . C a d a f a c t o r t e n d r a d o s l a s c u y a o r d e n a c i o n
s e r a u n a p e r m u t a c i o n d e f 1 2 3 4 g
P o r e j e m p l o
( 1 )
( 2 )
3 ; 2
2 0
; 1 2
2 ; 3
( 3 )
( 4 )
( 1 )
( 3 )
3 ; 2
1 3
5 2
2 ; 3
( 2 )
( 4 )
e t c .
E l s i g n o d e c a d a s u m a n d o s e r a e l s i g n o d e l a c o r r e s p o n d i e n t e p e r m u t a c i o n d e l a s .
P o r e j e m p l o , e l s i g n o d e l p r i m e r s u m a n d o a n t e r i o r e s e l s i g n o d e l a p e r m u t a c i o n
( 1 2 3 4 ) , q u e e s + , y e l d e l s e g u n d o , e l d e l a p e r m u t a c i o n ( 1 3 2 4 ) , q u e
e s ;
P a s e m o s p u e s a l c a l c u l o d e A
A = +
3 ; 2
2 0
; 1 2
2 3
;
3 ; 2
1 3
5 2
2 ; 3
+
3 ; 2
2 5
5 2
; 1 2
+
+
2 0
1 3
4 5
2 ; 3
;
2 0
2 5
4 5
; 1 2
+
1 3
2 5
4 5
5 2
=
= 4 ( ; 1 ) ; 1 1 ( ; 1 9 ) + 1 9 1 2 + 6 ( ; 2 2 ) ; 1 0 1 3 + ( ; 1 ) ( ; 1 7 ) = 1 8 8
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7 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
b )
A =
3 ; 2 4 5
2 0 5 2
1 3 ; 1 2
2 5 2 ; 3
= 3
0 5 2
3 ; 1 2
5 2 ; 3
; ( ; 2 )
2 5 2
1 ; 1 2
2 2 ; 3
+
+ 4
2 0 2
1 3 2
2 5 ; 3
; 5
2 0 5
1 3 ; 1
2 5 2
=
= 3 1 1 7 + 2 4 1 ; 4 4 0 ; 5 1 7 = 3 5 1 + 8 2 ; 1 6 0 ; 8 5 = 1 8 8
c ) S e g u i r e m o s u n m e t o d o q u e n o s p e r m i t e o b t e n e r e l m a x i m o n u m e r o d e c e r o s e n u n a
l n e a ( l a o c o l u m n a ) a b a s e d e s u m a r l e a d i c h a l n e a u n a c o m b i n a c i o n l i n e a l d e l a s
r e s t a n t e s . P o r e j e m p l o , c o m o e n l a s e g u n d a c o l u m n a h a y u n c e r o , e m p l e a m o s e s t a
c o l u m n a p a r a r e l l e n a r l a d e c e r o s
f i l a a ; !
f i l a b ; !
f i l a c ; !
f i l a d ; !
3 ; 2 4 5
2 0 5 2
1 3 ; 1 2
2 5 2 ; 3
s u b s t i t u i m o s l a l a c p o r c ; d ; a ( c o m b i n a c i o n l i n e a l d e l a s ) , e s d e c i r , l e s u m a m o s
a l a l a c l a s l a s a y d c a m b i a d a s d e s i g n o )
f i l a a ; !
f i l a b ; !
f i l a c ; !
f i l a d ; !
3 ; 2 4 5
2 0 5 2
; 4 0 ; 7 0
2 5 2 ; 3
s u b s t i t u i m o s , p o r e j e m p l o , l a l a d p o r d +
5
2
a , q u e d a n d o
3 ; 2 4 5
2 0 5 2
; 4 0 ; 7 0
1 9
2
0 1 2
1 9
2
D e s a r r o l l a n d o p u e s p o r l a s e g u n d a c o l u m n a t e n e m o s
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D e t e r m i n a n t e s 7 7
; ( ; 2 )
2 5 2
; 4 ; 7 0
1 9
2
1 2
1 9
2
= 2
1
2
2 5 2
; 4 ; 7 0
1 9 2 4 1 9
= 1 8 8
4 P r o b a r q u e
V
n
=
1 x
1
x
2
1
: : : x
n ; 1
1
1 x
2
x
2
2
: : : x
n ; 1
2
: : : : : : : : : : : : : : :
: : : : : : : : : : : : : : :
1 x
n
x
2
n
: : : x
n ; 1
n
=
Y
1 i < j n
( x
j
; x
i
)
p a r a n 2 ( E s e l l l a m a d o d e t e r m i n a n t e d e V a n d e r M o n d e ) .
S o l u c i o n :
V a m o s a p r o b a r l o p o r i n d u c c i o n
P a r a n = 2
V
2
=
1 x
1
1 x
2
= x
2
; x
1
S u p o n g a m o s q u e e s c i e r t o p a r a m = n ; 1 , v e a m o s q u e t a m b i e n l o e s p a r a n
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7 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
V
n
=
( a )
1 0 0 0 0
1 x
2
; x
1
x
2
2
; x
1
x
2
x
3
2
; x
2
2
x
1
: : : x
n ; 1
2
; x
n ; 2
2
x
1
1 x
3
; x
1
x
2
3
; x
1
x
3
x
3
3
; x
2
3
x
1
: : : x
n
3
; x
n ; 2
3
x
1
: : : : : : : : : : : : : : : : : :
1 x
n
; x
1
x
2
n
; x
n
x
1
x
3
n
; x
2
n
x
1
: : : x
n ; 1
n
; x
n ; 2
n
x
1
=
( b )
=
x
2
; x
1
x
2
2
; x
1
x
2
x
3
2
; x
2
2
x
1
: : : x
n ; 1
2
; x
n ; 2
2
x
1
x
3
; x
1
x
2
3
; x
1
x
3
x
3
3
; x
2
3
x
1
: : : x
n ; 1
3
; x
n ; 2
3
x
1
: : : : : : : : : : : : : : :
x
n
; x
1
x
2
n
; x
1
x
n ; 1
x
3
n
; x
2
n
x
1
: : : x
n ; 1
n
; x
n ; 2
n
x
1
=
( c )
= ( x
2
; x
1
) ( x
3
; x
1
) ( x
n
; x
1
)
1 x
2
x
2
2
: : : x
n ; 2
2
1 x
3
x
2
3
: : : x
n ; 2
3
: : : : : : : : : : : : : : :
1 x
n
x
2
n
: : : x
n ; 2
n
=
( d )
= ( x
2
; x
1
) ( x
3
; x
1
) ( x
n
; x
1
) V
n ; 1
=
Y
1 i < j n
( x
j
; x
i
)
( a ) r e s t a n d o a l a s e g u n d a c o l u m n a l a p r i m e r a m u l t i p l i c a d a p o r x
1
y a l a t e r c e r a
l a s e g u n d a p o r x
1
, . . . , y a l a n - s i m a l a ( n - 1 ) - s i m a p o r x
1
( b ) d e s a r r o l l a n d o p o r l a p r i m e r a l a .
( c ) s i u n a l a o c o l u m n a e s t a m u l t i p l i c a d a p o r u n e s c a l a r , e s t e s a l e f u e r a .
( d ) e l d e t e r m i n a n t e e s e l d e V a n d e r M o n d e d e o r d e n n - 1 .
5 C a l c u l a r
V =
1 1 1
b + c c + a a + b
b c a c a b
S o l u c i o n :
V =
( a )
1 0 0
b + c a ; b a ; c
b c a c ; b c a b ; b c
=
( b )
a ; b a ; c
c ( a ; b ) b ( a ; c )
=
= ( a ; b ) ( a ; c )
1 1
c b
= ( a ; b ) ( a ; c ) ( b ; c )
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D e t e r m i n a n t e s 7 9
( a ) r e s t a n d o l a p r i m e r a c o l u m n a a l a s e g u n d a y t e r c e r a .
( b ) d e s a r r o l l l a n d o p o r l a p r i m e r a l a .
6 C a l c u l a r
4 =
z ; z 0
0 z
2
; 1
1 z z + 1
s a b i e n d o q u e z 2 C e s t a l q u e z
5
= 1 y z 6= 1
S o l u c i o n :
4 = z z
2
( z + 1 ) + 0 z 0 + 1 ( ; z ) ( ; 1 ) ; 1 z
2
0 ; z z ( ; 1 ) ; 0 ( ; z ) ( z + 1 ) =
= z
4
+ z
3
+ z
2
+ z
A h o r a b i e n , p u e s t o q u e
0 = z
5
; 1 = ( z ; 1 ) ( z
4
+ z
3
+ z
2
+ z + 1 )
s e t i e n e
o
z ; 1 = 0
z
4
+ z
3
+ z
2
+ z + 1 = 0
p e r o z 6= 1
l u e g o z
4
+ z
3
+ z
2
+ z + 1 = 0 , d e d o n d e 4 = ; 1
7 C a l c u l a r e l d e t e r m i n a n t e d e o r d e n n s i g u i e n t e
A
n
=
2 1 0 0 0 0
1 2 1 0 0 0
0 1 2 1 0 0
: : : : : : : : : : : : : : : : : : : : :
0 0 0 0 1 2
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8 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S o l u c i o n :
D e s a r r o l l a n d o p o r l a p r i m e r a c o l u m n a t e n e m o s
A
n
= 2
2 1 0 0 0
1 2 1 0 0
: : : : : : : : : : : : : : : : : :
0 0 0 1 2
; 1
1 0 0 0 0
1 2 1 0 0
: : : : : : : : : : : : : : : : : :
0 0 0 1 2
=
=
( a )
2
2 1 0 0 0
1 2 1 0 0
: : : : : : : : : : : : : : : : : :
0 0 0 1 2
; 1 1
2 1 0 0 0
1 2 1 0 0
: : : : : : : : : : : : : : : : : :
0 0 0 1 2
=
= 2 A
n ; 1
; A
n ; 2
( a ) d e s a r r o l l a n d o e l s e g u n d o d e t e r m i n a n t e p o r l a p r i m e r a l a
L u e g o t e n e m o s l a r e l a c i o n d e r e c u r r e n c i a s i g u i e n t e
A
n
= 2 A
n ; 1
; A
n ; 2
q u e n o s p e r m i t i r a d e d u c i r e l v a l o r d e A
n
: t e n e m o s q u e p a r a n = 1 2 3 e s
A
1
= 2 A
2
= 3 A
3
= 4
S u p o n g a m o s p u e s q u e A
n ; 1
= n v e a m o s q u e A
n
= n + 1
A
n
= 2 A
n ; 1
; A
n ; 2
= 2 n ; ( n ; 1 ) = 2 n ; n + 1 = n + 1
l u e g o A
n
= n + 1
8 S i n e f e c t u a r e l d e s a r r o l l o , p r o b a r q u e
4 =
1 a b + c
1 b c + a
1 c a + b
= 0
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D e t e r m i n a n t e s 8 1
S o l u c i o n :
S a b e m o s q u e n o s e a l t e r a e l v a l o r d e u n d e t e r m i n a n t e s i a u n a l n e a l e s u m a m o s u n a
c o m b i n a c i o n l i n e a l d e l a s d e m a s . S u m a n d o a l a t e r c e r a c o l u m n a l a s e g u n d a , n o s q u e d a
4 =
1 a a + b + c
1 b b + c + a
1 c c + a + b
= ( a + b + c )
1 a 1
1 b 1
1 c 1
=
( a )
0
( a ) o b s e r v a n d o q u e h a y d o s c o l u m n a s i g u a l e s
9 C a l c u l a r l a s r a c e s d e l a e c u a c i o n
4
n
=
1 + x 1 1
1 1 + x : : : 1
: : : : : : : : : : : :
1 1 1 + x
S o l u c i o n :
S u m a n d o a l a p r i m e r a c o l u m n a t o d a s l a s d e m a s t e n e m o s
4
n
=
n + x 1 1
n + x 1 + x : : : 1
: : : : : : : : : : : :
n + x 1 1 + x
= ( n + x )
1 1 1
1 1 + x : : : 1
: : : : : : : : : : : :
1 1 1 + x
=
=
( a )
( n + x )
1 0 0
1 x : : : 0
: : : : : : : : : : : :
1 0 : : : x
= ( n + x ) x
n ; 1
= 0
( a ) r e s t a n d o a c a d a c o l u m n a , a p a r t i r d e l a s e g u n d a , l a p r i m e r a c o l u m n a
L u e g o l a s r a c e s s o n x = ; n y x = 0 d e m u l t i p l i c i d a d n - 1 .
1 0 S a b i e n d o q u e 1 8 8 8 7 , 3 9 8 6 5 , 5 8 7 5 2 , 6 4 8 7 2 , 9 6 5 2 6 s o n d i v i s i b l e s p o r 1 7 , d e m o s t r a r
q u e D e s t a m b i e n m u l t i p l o d e 1 7 , s i e n d o D e l d e t e r m i n a n t e s i g u i e n t e
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8 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
D =
1 8 8 8 7
3 9 8 6 5
5 8 7 5 2
6 4 8 7 2
9 6 5 2 6
s i n c a l c u l a r e l v a l o r d e l d e t e r m i n a n t e .
S o l u c i o n :
S a b e m o s q u e
1 8 8 8 7 = 1 7 a
3 9 8 6 5 = 1 7 b
5 8 7 5 2 = 1 7 c
6 4 8 7 2 = 1 7 d
9 6 5 2 6 = 1 7 e
9
>
>
>
>
>
=
>
>
>
>
>
c o n a b c d 2 Z
D =
1
1 0
4
1 1 0
4
8 8 8 7
3 1 0
4
9 8 6 5
5 1 0
4
8 7 5 2
6 1 0
4
4 8 7 2
9 1 0
4
6 5 2 6
=
=
1
1 0
4
1 1 0
4
+ 8 1 0
3
+ 8 1 0
2
+ 8 1 0 + 7 8 8 8 7
3 1 0
4
+ 9 1 0
3
+ 8 1 0
2
+ 6 1 0 + 5 9 8 6 5
5 1 0
4
+ 8 1 0
3
+ 7 1 0
2
+ 5 1 0 + 2 8 7 5 2
6 1 0
4
+ 4 1 0
3
+ 8 1 0
2
+ 7 1 0 + 2 4 8 7 2
9 1 0
4
+ 6 1 0
3
+ 5 1 0
2
+ 2 1 0 + 6 6 5 2 6
=
=
1
1 0
4
1 8 8 8 7 8 8 8 7
3 9 8 6 5 9 8 6 5
5 8 7 5 2 8 7 5 2
6 4 8 7 2 4 8 7 2
9 6 5 2 6 6 5 2 6
=
1
1 0
4
1 7 a 8 8 8 7
1 7 b 9 8 6 5
1 7 c 8 7 5 2
1 7 d 4 8 7 2
1 7 e 6 5 2 6
=
=
1 7
1 0
4
a 8 8 8 7
b 9 8 6 5
c 8 7 5 2
d 4 8 7 2
e 6 5 2 6
=
1 7
1 0
4
D
0
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D e t e r m i n a n t e s 8 3
D =
1 7
1 0
4
D
0
p u e s t o q u e m c d ( 1 7 1 0
4
) = 1 y D D
0
2 Z , s e t i e n e q u e
D
0
= 1 0
4
h y p o r t a n t o D = 1 7 h
1 1 D e t e r m i n a r l a i n v e r s a d e l a m a t r i z A = ( a
i j
)
A =
0
@
1 2 2
2 1 2
2 2 1
1
A
S o l u c i o n :
A
; 1
=
1
d e t A
( A
i j
)
S i e n d o
A
j i
= ( ; 1 )
i + j
d e t
0
B
B
B
B
B
@
a
1 1
a
1 2
: : : a
1 j ; 1
a
1 j + 1
: : : a
1 n
: : : : : : : : : : : : : : : : : : : : :
a
i ; 1 1
a
i ; 1 2
: : : : : : : : : : : : a
i ; 1 n
a
i + 1 1
a
i + 1 2
: : : : : : : : : : : : a
i + 1 n
: : : : : : : : : : : : : : : : : : : : :
a
n 1
a
n 2
: : : : : : : : : : : : a
n n
1
C
C
C
C
C
A
N o t a : A
i j
= A
t
j i
L u e g o , d e t A = 5
A
1 1
= ( ; 1 )
2
1 2
2 1
= ; 3 A
1 2
= ( ; 1 )
3
2 2
2 1
= 2
A
1 3
= ( ; 1 )
4
2 2
1 2
= 2 A
2 1
= ( ; 1 )
3
2 2
2 1
= 2
A
2 2
= ( ; 1 )
4
1 2
2 1
= ; 3 A
2 3
= ( ; 1 )
5
1 2
2 2
= 2
A
3 1
= ( ; 1 )
4
2 1
2 2
= 2 A
3 2
= ( ; 1 )
5
1 2
2 2
= 2
A
3 3
= ( ; 1 )
6
1 2
2 1
= ; 3
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8 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
p o r l o q u e l a m a t r i z i n v e r s a e s :
A
; 1
=
1
5
; 3 2 2
2 ; 3 2
2 2 ; 3
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D i a g o n a l i z a c i o n d e e n d o m o r s m o s 8 5
C a p t u l o 6 D i a g o n a l i z a c i o n d e e n d o m o r s m o s
1 S e a E u n R - e s p a c i o v e c t o r i a l y f u n e n d o m o r s m o d e E c u y a m a t r i z e n u n a
d e t e r m i n a d a b a s e f u
1
u
2
u
3
g e s
A =
0
@
1 2 1
0 1 2
3 1 0
1
A
H a l l a r e l p o l i n o m i o c a r a c t e r s t i c o Q ( t )
Q ( t ) =
1 ; t 2 1
0 1 ; t 2
3 1 0 ; t
=
= ; t
3
+ ( t r A ) t
2
; ( A
1 1
+ A
2 2
+ A
3 3
) t + d e t A =
= ; t
3
+ 2 t
2
+ 4 t + 7
N o t a A
i i
e s e l d e t e r m i n a n t e d e l m e n o r a d j u n t o a l e l e m e n t o a
i i
d e l a m a t r i z A
2 D e t e r m i n a r e l p o l i n o m i o c a r a c t e r s t i c o d e l a m a t r i z
A =
0
B
@
a
2
a b a b b
2
a b a
2
b
2
a b
a b b
2
a
2
a b
b
2
a b a b a
2
1
C
A
d a n d o s u s r a c e s .
S o l u c i o n :
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8 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
d e t ( A ; t I ) =
( a + b )
2
; t a b a b b
2
( a + b )
2
; t a
2
; t b
2
a b
( a + b )
2
; t b
2
a
2
; t a b
( a + b )
2
; t a b a b a
2
; t
=
= ( ( a + b )
2
; t )
1 a b a b b
2
1 a
2
; t b
2
a b
1 b
2
a
2
; t a b
1 a b a b a
2
; t
=
= ( ( a + b )
2
; t )
1 a b a b b
2
0 a
2
; a b ; t b
2
; a b a b ; b
2
0 b
2
; a b a
2
; a b ; t a b ; b
2
0 0 0 a
2
; b
2
; t
=
= ( ( a + b )
2
; t )
a
2
; a b ; t b
2
; a b a b ; b
2
b
2
; a b a
2
; a b ; t a b ; b
2
0 0 a
2
; b
2
; t
=
= ( ( a + b )
2
; t ) ( a
2
; b
2
; t )
a
2
; a b ; t b
2
; a b
b
2
; a b a
2
; a b ; t
=
= ( ( a + b )
2
; t ) ( a
2
; b
2
; t )
a
2
+ b
2
; 2 a b ; t b
2
; a b
a
2
+ b
2
; 2 a b ; t a
2
; a b ; t
=
= ( ( a + b )
2
) ( a
2
; b
2
; t ) ( ( a ; b )
2
; t )
1 b
2
; a b
1 a
2
; a b ; t
=
= ( ( a + b )
2
; t ) ( a
2
; b
2
; t ) ( ( a ; b )
2
; t )
1 b
2
; a b
0 a
2
; b
2
; t
=
= ( ( a + b )
2
; t ) ( a
2
; b
2
; t )
2
( ( a ; b )
2
; t )
y p o r l o t a n t o , l a s r a c e s s o n ( a + b )
2
( a ; b )
2
( a
2
; b
2
) , y e s t a u l t i m a d e m u l t i -
p l i c i d a d d o s .
3 S i A 2 M
n
( K ) e s u n a m a t r i z i n v e r s i b l e , d e m o s t r a r q u e A B y B A t i e n e n l o s
m i s m o s v a l o r e s p r o p i o s ( s i e n d o K u n c u e r p o c o n m u t a t i v o ) .
S o l u c i o n :
E n e f e c t o ,
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D i a g o n a l i z a c i o n d e e n d o m o r s m o s 8 7
d e t ( A B ; I ) = d e t ( A B A A
; 1
; I A A
; 1
) = d e t ( A B A A
; 1
; A I A
; 1
) =
= d e t ( A ( B A ; I ) A
; 1
) = d e t A d e t ( B A ; I ) d e t A
; 1
=
= d e t ( B A ; I )
4 D e m o s t r a r q u e s i l a m a t r i z A 2 M
n
( K ) v e r i c a A
m
= 0 , e l u n i c o v a l o r p r o p i o
p o s i b l e d e A e s e l c e r o ( d o n d e K e s u n c u e r p o c o n m u t a t i v o ) .
S o l u c i o n :
S u p o n g a m o s l o c o n t r a r i o , e s d e c i r , s u p o n g a m o s q u e e x i s t e 6= 0 q u e s e a v a l o r p r o p i o
d e l a m a t r i z A , l o q u e e q u i v a l e a q u e s e a u n v a l o r p r o p i o d e l e n d o m o r s m o f d e l
e s p a c i o v e c t o r i a l K
n
c u y a m a t r i z e n d e t e r m i n a d a b a s e e s A y e s t o s i g n i c a q u e e x i s t e
u n v e c t o r v 2 K
n
v 6= 0 t a l q u e f ( v ) = v 6= 0
Y a p l i c a n d o f a a m b o s m i e m b r o s d e l a i g u a l d a d s e t i e n e
f
2
( v ) = f ( v ) = f ( v ) =
2
v
l u e g o
2
e s u n v a l o r p r o p i o n o n u l o d e f
2
y p o r t a n t o d e A
2
i n d u c t i v a m e n t e t e n e m o s q u e f
m
=
m
v 6= 0 , e n e f e c t o :
S a b e m o s q u e e s c i e r t o p a r a m = 1 2 s u p o n g a m o s q u e l o e s p a r a m ; 1 v e a m o s q u e
l o e s p a r a m
f ( f
m ; 1
v ) = f (
m ; 1
v ) =
m ; 1
f ( v ) =
m ; 1
v =
m
v
P o r l o t a n t o t e n e m o s q u e
m
e s v a l o r p r o p i o n o n u l o d e l a m a t r i z A
m
= 0 l o c u a l
e s a b s u r d o .
5 S e d e n e f : R
3
; ! R
3
p o r f ( x
1
x
2
x
3
) = ( x
1
x
2
0 ) 8 ( x
1
x
2
x
3
) 2 R
3
C o m p r o b a r q u e f e s l i n e a l y h a l l a r s u m a t r i z e n l a b a s e n a t u r a l . H a l l a r e l p o l i n o m i o
c a r a c t e r s t i c o y l o s v a l o r e s p r o p i o s . > E s f d i a g o n a l i z a b l e ?
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8 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S o l u c i o n :
V e a m o s l a l i n e a l i d a d :
8 v = ( x
1
x
2
x
3
) w = ( y
1
y
2
y
3
) 2 R
3
y 8 2 R s e t i e n e :
f ( v + w ) = ( x
1
+ y
1
x
2
+ y
2
0 ) = ( x
1
x
2
0 ) + ( y
1
y
2
0 ) = f ( v ) + f ( w )
f ( v ) = f ( x
1
x
2
x
3
) = ( x
1
x
2
0 ) = ( x
1
x
2
0 ) = f ( v )
D e t e r m i n e m o s l a m a t r i z d e l a a p l i c a c i o n f
f ( e
1
) = e
1
f ( e
2
) = e
2
f ( e
3
) = 0
9
>
=
>
( 1 )
l u e g o l a m a t r i z d e f e n l a b a s e n a t u r a l e s :
A =
0
@
1 0 0
0 1 0
0 0 0
1
A
( 2 )
E l p o l i n o m i o c a r a c t e r s t i c o e s :
d e t ( f ; t I ) = ( 1 ; t )
2
( ; t ) = ; t
3
+ 2 t
2
; t
( o b v i o y a q u e l a m a t r i z e s d i a g o n a l )
L o s v a l o r e s p r o p i o s s o n l o s v a l o r e s 2 R t a l e s q u e 9 v 2 R
3
v 6= 0 c o n f ( v ) = v y
e s t o s v a l o r e s s o n l a s r a c e s d e l p o l i n o m i o c a r a c t e r s t i c o :
( 1 ; t )
2
( ; t ) = 0 t = 1 d o b l e t = 0
f d i a g o n a l i z a p u e s t o q u e d i m K e r ( f ; I ) = 2 .
E n ( 1 ) o b s e r v a m o s y a q u e l a b a s e n a t u r a l e s l a b a s e d e v e c t o r e s p r o p i o s , ( f e s l a
\ p r o y e c c i o n o r t o g o n a l " s o b r e e l p l a n o h o r i z o n t a l X Y ) , y e n ( 2 ) v e m o s q u e l a m a t r i z e s
d i a g o n a l .
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D i a g o n a l i z a c i o n d e e n d o m o r s m o s 8 9
6 D i a g o n a l i z a r l a m a t r i z
A =
0
@
; 2 1 2 8
2 0 ; 3 ; 8
; 6 0 6 2 3
1
A
h a l l a n d o u n a b a s e d e v e c t o r e s p r o p i o s d e f ( e n d o m o r s m o d e R
3
c u y a
m a t r i z e n l a b a s e n a t u r a l e s A )
S o l u c i o n :
B u s q u e m o s e l p o l i n o m i o c a r a c t e r s t i c o d e A :
d e t ( A ; t I ) = ; t
3
; t
2
+ t + 1 = ; ( t + 1 )
2
( t ; 1 )
p o r l o t a n t o
R
3
= K e r ( f + I )
2
K e r ( f ; I )
d i m K e r ( f + I ) = 3 ; r a n g o
0
@
; 2 0 2 8
2 0 ; 2 ; 8
; 6 0 6 2 4
1
A
= 3 ; 1 = 2
l u e g o A d i a g o n a l i z a y
D =
0
@
; 1 0 0
0 ; 1 0
0 0 1
1
A
l a n u e v a b a s e s e r a f v
1
v
2
v
3
g c o n v
1
v
2
2 K e r ( f + I ) y v
3
2 K e r ( f ; I )
K e r ( f + I ) = f ( x y z ) =
0
@
; 2 0 2 8
2 0 ; 2 ; 8
; 6 0 6 2 4
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
g =
= f ( x y z ) = ; 1 0 x + y + 4 z = 0 g
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9 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
s u b e s p a c i o d e d i m e n s i o n d o s , d e l q u e s e l e c c i o n a m o s u n a b a s e
v
1
= ( 2 0 5 ) v
2
= ( 1 2 2 )
k e r ( f ; I ) = f ( x y z ) =
0
@
; 2 2 2 8
2 0 ; 4 ; 8
; 6 0 6 2 2
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
g =
= f ( x y z ) = ; 1 1 x + y + 4 z = 0 1 0 x ; 2 y ; 4 z = 0 g
s u b e s p a c i o d e d i m e n s i o n u n o d e l q u e s e l e c c i o n a m o s u n a b a s e
v
3
= ( ; 1 1 ; 3 )
7 E s t u d i a r l a d i a g o n a l i z a c i o n , s e g u n l o s d i s t i n t o s v a l o r e s d e 2 R , d e l a m a t r i z
A =
0
@
1 ; ; ;
+ 1 ; 1
0 0 2
1
A
d a n d o e n e l c a s o e n q u e e l l o s e a p o s i b l e u n a m a t r i z S t a l q u e S
; 1
A S s e a
d i a g o n a l .
S o l u c i o n :
B u s q u e m o s e l p o l i n o m i o c a r a c t e r s t i c o :
d e t ( A ; t I ) = ; ( t ; 1 )
2
( t ; 2 )
l u e g o l o s v a l o r e s p r o p i o s d e A s o n t
1
= 1 d o b l e y t
2
= 2
P a r a q u e A d i a g o n a l i c e , h a d e v e r i c a r s e :
1 d i m k e r ( A ; t
i
I ) = m u l t i p l i c i d a d d e l a r a z t
i
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D i a g o n a l i z a c i o n d e e n d o m o r s m o s 9 1
E s t u d i e m o s p u e s e l c a s o t
1
= 1
d i m K e r ( A ; I ) = 3 ; r a n g o
0
@
; ; ;
; 1
0 0 1
1
A
=
3 ; 1 = 2 p a r a = 0
3 ; 2 = 1 p a r a 6= 0
l u e g o , s o l o d i a g o n a l i z a p a r a = 0 . S e a p u e s = 0 y b u s q u e m o s l a m a t r i z S ( m a t r i z
d e l o s v e c t o r e s p r o p i o s ) .
S e a n f v
1
v
2
g b a s e d e K e r ( A ; I )
0
@
0 0 0
0 0 ; 1
0 0 1
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
)
; z = 0
z = 0
s e a n p u e s v
1
= ( 1 0 0 ) v
2
= ( 0 1 0 )
y f v
3
g b a s e d e K e r ( A + I )
0
@
; 1 0 0
0 ; 1 ; 1
0 0 0
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
)
; x = 0
; y ; z = 0
S e a p u e s v
3
= ( 0 1 ; 1 )
l u e g o S =
0
@
1 0 0
0 1 1
0 0 ; 1
1
A
y e n e f e c t o , s e t i e n e q u e D = S
; 1
A S
D =
0
@
1 0 0
0 1 0
0 0 2
1
A
=
0
@
1 0 0
0 1 1
0 0 ; 1
1
A
0
@
1 0 0
0 1 ; 1
0 0 2
1
A
0
@
1 0 0
0 1 1
0 0 ; 1
1
A
N o t a : E n e s t e c a s o S = S
; 1
8 S e a f : R
3
; ! R
3
u n e n d o m o r s m o d i a g o n a l i z a b l e q u e a d m i t e p o r v e c t o r e s
p r o p i o s a l o s v
1
= ( ; 1 2 2 ) v
2
= ( 2 2 ; 1 ) v
3
= ( 2 ; 1 2 ) y s a b e m o s q u e
f ( 5 2 5 ) = ( 0 0 7 ) . H a l l a r l o s v a l o r e s p r o p i o s d e f
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9 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S o l u c i o n :
P u e s t o q u e f ( v
i
) =
i
v
i
e x p r e s a r e m o s l o s v e c t o r e s ( 5 2 5 ) y ( 0 0 7 ) e n l a b a s e
f o r m a d a p o r l o s v e c t o r e s p r o p i o s d e f y a p l i c a m o s f , a l p r i m e r o
S =
0
@
; 1 2 2
2 2 ; 1
2 ; 1 2
1
A
S
; 1
=
1
9
0
@
; 1 2 2
2 2 ; 1
2 ; 1 2
1
A
p o r l o t a n t o
1
9
0
@
; 1 2 2
2 2 ; 1
2 ; 1 2
1
A
0
@
5
2
5
1
A
=
0
@
1
1
2
1
A
1
9
0
@
; 1 2 2
2 2 ; 1
2 ; 1 2
1
A
0
@
0
0
7
1
A
=
1
9
0
@
1 4
; 7
1 4
1
A
l u e g o
f ( 1 1 2 ) = f ( v
1
+ v
2
+ 2 v
3
) = f ( v
1
) + f ( v
2
) + 2 f ( v
3
) =
=
1
v
1
+
2
v
2
+ 2
3
v
3
=
1 4
9
v
1
;
7
9
v
2
+
1 4
9
v
3
Y
1
=
1 4
9
2
= ;
7
9
3
=
7
9
y a q u e f v
i
g e s u n a b a s e d e l e s p a c i o y l a e x p r e s i o n
d e u n v e c t o r e n u n a d e t e r m i n a d a b a s e e s u n i c a .
9 S e a f u n e n d o m o r s m o d e R
n
. P r o b a r q u e s i 2 R e s u n v a l o r p r o p i o d e
f e n t o n c e s
p
e s u n v a l o r p r o p i o d e f
p
8 p 2 N y l o s s u b e s p a c i o s p r o p i o s r e s p e c t i v o s
E E
p
s o n t a l e s q u e E E
p
. D a r u n e j e m p l o e n e l q u e E 6= E
p
S o l u c i o n :
S e a u n v a l o r p r o p i o , e x i s t e p u e s u n v e c t o r x 2 R
n
x 6= 0 t a l q u e f ( x ) = x
l u e g o
f
2
( x ) = f ( f ( x ) ) = f ( x ) = f ( x ) =
2
x
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D i a g o n a l i z a c i o n d e e n d o m o r s m o s 9 3
e s d e c i r
2
e s v a l o r p r o p i o d e f
2
d e v e c t o r p r o p i o x
S u p o n g a m o s q u e h e m o s p r o b a d o q u e t a m b i e n
p ; 1
e s v a l o r p r o p i o d e f
p ; 1
d e v e c t o r
p r o p i o x , e n t o n c e s
f
p
( x ) = f ( f
p ; 1
( x ) ) = f (
p ; 1
x ) =
p ; 1
f ( x ) =
p ; 1
x =
p
x
l u e g o
p
e s v a l o r p r o p i o d e f
p
d e v e c t o r p r o p i o x , y o b v i a m e n t e , p a r a t o d o v e c t o r
x p r o p i o d e v a l o r p r o p i o d e f , p o d e m o s a p l i c a r l e e l r a z o n a m i e n t o a n t e r i o r , y s e
t i e n e
E E
p
V e a m o s q u e l a i g u a l d a d e n g e n e r a l e s f a l s a s e a f 2 E n d ( R
3
) t a l q u e s u m a t r i z e n l a
b a s e n a t u r a l e s
0
@
0 0 0
1 0 0
0 1 0
1
A
t e n e m o s q u e f e
3
g e s e l s u b e s p a c i o d e v e c t o r e s p r o p i o s d e v a l o r p r o p i o c e r o . S i n e m -
b a r g o f
2
e s t a l q u e s u m a t r i z e n l a b a s e n a t u r a l e s
0
@
0 0 0
0 0 0
1 0 0
1
A
y f e
2
e
3
g e s e l s u b e s p a c i o d e v e c t o r e s p r o p i o s d e v a l o r p r o p i o c e r o y c l a r a m e n t e
E 6 E
2
1 0 S e a A l a m a t r i z
0
@
1 2 2
2 1 2
2 2 1
1
A
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9 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
D e t e r m i n a r A
n
, p a r a t o d o n 2 N
S o l u c i o n :
S i D e s u n a m a t r i z d i a g o n a l D = (
i i
) s e t i e n e c l a r a m e n t e D
n
= (
n
i i
)
S i A e s d i a g o n a l i z a b l e , e x i s t e S t a l q u e D = S
; 1
A S y
D
n
= ( S
; 1
A S )
n
= S
; 1
A
n
S l u e g o A
n
= S D
n
S
; 1
V e a m o s s i A e s d i a g o n a l i z a b l e :
d e t ( A ; t I ) = ; ( t + 1 )
2
( t + 5 )
l o s v a l o r e s p r o p i o s s o n t = ; 1 d o b l e y t = 5
d i m K e r ( A + I ) = 3 ; r a n g o
0
@
2 2 2
2 2 2
2 2 2
1
A
= 3 ; 1 = 2
l u e g o A d i a g o n a l i z a .
B u s q u e m o s l a m a t r i z S :
v
1
v
2
2 K e r ( A + I )
0
@
2 2 2
2 2 2
2 2 2
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
) x + y + z = 0 )
v
1
= (
p
2
2
;
p
2
2
0 )
v
2
= (
p
6
6
p
6
6
;
p
6
3
)
9
>
>
=
>
>
v
3
2 K e r ( A ; 5 I )
0
@
; 4 2 2
2 ; 4 2
2 2 ; 4
1
A
0
@
x
y
z
1
A
)
; 2 x + y + z = 0
x ; 2 y + z = 0
) v
3
= (
p
3
3
p
3
3
p
3
3
)
l u e g o
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D i a g o n a l i z a c i o n d e e n d o m o r s m o s 9 5
D =
0
B
B
B
B
@
p
2
2
;
p
2
2
0
p
6
6
p
6
6
;
p
6
3
p
3
3
p
3
3
p
3
3
1
C
C
C
C
A
0
B
B
B
@
1 2 2
2 1 2
2 2 1
1
C
C
C
A
0
B
B
B
B
@
p
2
2
p
6
6
p
3
3
;
p
2
2
p
6
6
p
3
3
0 ;
p
6
3
p
3
3
1
C
C
C
C
A
=
=
0
@
; 1 0 0
0 ; 1 0
0 0 5
1
A
Y
D
n
=
0
@
( ; 1 )
n
( ; 1 )
n
5
n
1
A
F i n a l m e n t e
A
n
= S D
n
S
; 1
=
=
0
B
B
B
@
2
3
( ; 1 )
n
+
1
3
5
n
;
1
3
( ; 1 )
n
+
1
3
5
n
1
3
( ; 1 )
n + 1
+
1
3
5
n
;
1
3
( ; 1 )
n
+
1
3
5
n
2
3
( ; 1 )
n
+
1
3
5
n
( ; 1 )
n
1
3
+
1
3
5
n
1
3
( ; 1 )
n + 1
+
1
3
5
n
1
3
( ; 1 )
n
+
1
3
5
n
( ; 1 )
n
2
3
+
1
3
5
n
1
C
C
C
A
1 1 a ) S e a A 2 M
n n
( C ) . P r o b a r q u e A y A
t
t i e n e n e l m i s m o p o l i n o m i o c a r a c -
t e r s t i c o .
b ) S e a n A 2 M
n n
( C ) , B 2 M
m m
( C ) , C 2 M
n m
( C ) , t a l e s q u e A C ; C B = C
P r o b a r q u e
8 p 2 N A
p
C = C ( I
m
+ B )
p
c ) S e a E = M
n m
( C ) y f u n e n d o m o r s m o d e E d e n i d o d e l a f o r m a f ( X ) =
A X ; X B c o n A 2 M
n n
( C ) y B 2 M
m m
( C ) j a s . P r o b a r q u e 2 C e s u n v a l o r
p r o p i o d e f s i y s o l o s i =
i
;
j
c o n
i
y
j
v a l o r e s p r o p i o s d e l o s e n d o m o r s m o s
d e C
n
y C
m
a s o c i a d o s a l a s m a t r i c e s A y B r e s p e c t i v a m e n t e .
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9 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S o l u c i o n :
a ) E n e f e c t o : d e t ( A ; I ) = d e t ( A ; I )
t
= d e t ( A
t
; I )
b ) V e a m o s l o p o r i n d u c c i o n r e s p e c t o a p . S e v e r i c a c l a r a m e n t e p a r a p = 1 : d e
A C ; C B = C t e n e m o s
A C = C + C B = C ( I
m
) + C B = C ( I
m
+ B )
s u p o n g a m o s a h o r a q u e e s c i e r t o p a r a p y v e a m o s q u e l o e s p a r a p + 1
A
p + 1
C = A A
p
C = A ( C ( I
m
+ B )
p
) = ( A C ) ( I
m
+ B )
p
=
= ( C ( I
m
+ B ) ) ( I
m
+ B )
p
= C ( I
m
+ B )
p + 1
c ) S e a f
1
: : :
n
g e l c o n j u n t o d e v a l o r e s p r o p i o s d e A y f
1
: : :
j
g e l c o n j u n t o d e
v a l o r e s p r o p i o s d e B
S e a
i
u n v a l o r p r o p i o d e A , e x i s t e v v e c t o r c o l u m n a d e C
n
; f 0 g t a l q u e A v =
i
v
S e a
j
u n v a l o r p r o p i o d e B e n t o n c e s , p o r a , e s t a m b i e n v a l o r p r o p i o d e B
t
, p o r l o q u e
e x i s t e w v e c t o r c o l u m n a d e C
m
; f 0 g t a l q u e B
t
w =
j
w y p o r t a n t o w
t
B =
j
w
t
S e a a h o r a X = v w
t
2 M
n m
( C )
f ( X ) = A X ; X B = A v w
t
; v w
t
B =
=
i
v w
t
;
j
v w
t
= (
i
;
j
) v w
t
= (
i
;
j
) X
p o r l o q u e
i
;
j
e s u n v a l o r p r o p i o d e f
R e c p r o c a m e n t e
S e a X 6= 0 u n v e c t o r p r o p i o d e f d e v a l o r p r o p i o ( f ( X ) = X )
S e a P ( t ) = ( ; 1 )
n
( t ;
1
) ( t ;
n
) ( l o s
i
n o n e c e s a r i a m e n t e d i s t i n t o s ) e l p o l i n o m i o
c a r a c t e r s t i c o d e A ( r e c u e r d e s e q u e C e s a l g e b r a i c a m e n t e c e r r a d o p o r l o q u e t o d o s l o s
f a c t o r e s p r i m o s d e P ( t ) s o n d e g r a d o 1 ) .
P o r e l t e o r e m a d e C a y l e y - H a m i l t o n P ( A ) = 0 p o r l o q u e P ( A ) X = 0 . A h o r a b i e n ,
p o r b , P ( A ) X = X P ( I
m
+ B )
T e n e m o s p u e s
0 = X P ( I
m
+ B ) = X ( I
m
;
1
I
m
) ( I
m
;
n
I
m
) =
= X ( ( ;
1
) I
m
+ B ) ( ( ;
n
) I
m
+ B ) = X C C 2 M
m m
( C )
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D i a g o n a l i z a c i o n d e e n d o m o r s m o s 9 7
X e s u n a m a t r i z n o n u l a p o r l o q u e C n o p u e d e s e r d e r a n g o m a x i m o :
d e t C =
n
Y
i = 1
( d e t ( ( ;
i
) I
m
+ B ) ) = 0
e x i s t e , p u e s , a l g u n i p a r a e l c u a l d e t ( ( ;
i
) I
m
+ B ) = 0 e s d e c i r ; ( ;
i
) e s u n
v a l o r p r o p i o d e B p o r l o q u e e x i s t e a l g u n j p a r a e l c u a l
j
= ; ( ;
i
) , e s d e c i r
=
i
;
j
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9 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
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F o r m a r e d u c i d a d e J o r d a n 9 9
C a p t u l o 7 F o r m a r e d u c i d a d e J o r d a n
1 H a l l a r e l p o l i n o m i o a n u l a d o r d e l a m a t r i z :
A =
0
@
2 0 0
0 2 0
1 2 2
1
A
s i n u t i l i z a r e l p o l i n o m i o c a r a c t e r s t i c o .
H a y q u e b u s c a r u n p o l i n o m i o P ( t ) = a
0
+ a
1
t + + a
r
t
r
t a l q u e P ( A ) =
a
0
I + a
1
A + + a
r
A
r
s e a l a m a t r i z n u l a .
E m p e c e m o s s u p o n i e n d o q u e P ( t ) e s u n p o l i n o m i o d e p r i m e r g r a d o : a
0
+ a
1
t y p l a n t e e m o s
P ( A ) = a
0
I + a
1
A = 0 e s d e c i r
0
@
a
0
0 0
0 a
0
0
0 0 a
0
1
A
+
0
@
2 a
1
0 0
0 2 a
1
0
a
1
2 a
1
2 a
1
1
A
=
0
@
0 0 0
0 0 0
0 0 0
1
A
l o c u a l i m p l i c a :
a
0
+ 2 a
1
= 0
a
1
= 0
o s e a a
0
= a
1
= 0
E s t o n o s d i c e q u e n o p u e d e f o r m a r s e l a m a t r i z n u l a p o r c o m b i n a c i o n l i n e a l n o n u l a d e
I y A p o r l o q u e P ( t ) n o p u e d e s e r d e p r i m e r g r a d o .
I n t e n t e m o s a h o r a c o n u n p o l i n o m i o d e s e g u n d o g r a d o P ( t ) = a
0
+ a
1
t + a
2
t
2
y
c a l c u l e m o s A
2
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1 0 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
A
2
=
0
@
4 0 0
0 4 0
4 8 4
1
A
a
0
0
@
1 0 0
0 1 0
0 0 1
1
A
+ a
1
0
@
2 0 0
0 2 0
1 2 2
1
A
+ a
2
0
@
4 0 0
0 4 0
4 8 4
1
A
=
0
@
0 0 0
0 0 0
0 0 0
1
A
l o c u a l i m p l i c a
a
0
+ 2 a
1
+ 4 a
2
= 0
a
1
+ 4 a
2
= 0
) a
0
= ; a
1
= 4 a
2
P u e d e t o m a r s e a
0
= 4 a
1
= ; 4 a
2
= 1 , p a r a t e n e r P ( t ) n o r m a l i z a d o , p o r l o q u e
P ( t ) = t
2
; 4 t + 4 = ( t ; 2 )
2
2 S a b i e n d o q u e u n e n d o m o r s m o f d e R
1 1
t i e n e ( t + 1 )
2
( t ; 4 )
3
( t + 2 )
6
c o m o p o l i -
n o m i o c a r a c t e r s t i c o y ( t + 1 )
2
( t ; 4 ) ( t + 2 )
3
c o m o p o l i n o m i o a n u l a d o r . > C u a l e s s o n
s u s p o s i b l e s f o r m a s d e J o r d a n ?
S o l u c i o n :
D e :
Q ( t ) = ( t + 1 )
2
( t ; 4 )
3
( t + 2 )
6
P ( t ) = ( t + 1 )
2
( t ; 4 ) ( t + 2 )
3
S e t i e n e l a 1
a
d e s c o m p o s i c i o n :
R
1 1
= K e r ( f + I )
2
K e r ( f ; 4 I ) K e r ( f + 2 I )
3
= E
1
E
2
E
3
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F o r m a r e d u c i d a d e J o r d a n 1 0 1
P a s e m o s a l a d e s c o m p o s i c i o n d e l o s E
i
e n m o n o g e n o s e l p o l i n o m i o a n u l a d o r d e E
1
e s
( t + 1 )
2
, l u e g o l a d i m e n s i o n d e l m o n o g e n o m a y o r e s 2 y c o i n c i d e c o n e l p o l i n o m i o
c a r a c t e r s t i c o ( t + 1 )
2
, q u e n o s d i c e q u e d i m E
1
= 2 , l u e g o e n E
1
h a y u n s o l o
m o n o g e n o E
1
= E
1 1
, y l a m a t r i z d e f r e s t r i n g i d a a E
1
e s :
; 1 0
1 ; 1
E l p o l i n o m i o a n u l a d o r d e E
2
e s ( t ; 4 ) , l u e g o f r e s t r i n g i d o a E
2
d i a g o n a l i z a , y a
q u e l a d i m e n s i o n d e l m o n o g e n o m a y o r e s 1 , y p u e s t o q u e e l p o l i n o m i o c a r a c t e r s t i c o
r e s t r i n g i d o a E
2
e s ( t ; 4 )
3
, s e t i e n e q u e l a d i m e n s i o n d e E
2
e s 3 , p o r l o q u e h a y
t r e s m o n o g e n o s e n E
2
d e d i m e n s i o n 1 y e s E
2
= E
2 1
E
2 2
E
2 3
y l a m a t r i z d e
f r e s t r i n g i d a a E
2
e s :
0
@
4 0 0
0 4 0
0 0 4
1
A
E l p o l i n o m i o a n u l a d o r d e E
3
e s ( t + 2 )
3
, l u e g o l a d i m e n s i o n d e l m o n o g e n o m a y o r e s
3 y p u e s t o q u e e l p o l i n o m i o c a r a c t e r s t i c o d e E
3
e s ( t + 1 )
6
, l a d i m e n s i o n d e E
3
e s
6 , l u e g o n o t e n e m o s u n v o c a m e n t e d e t e r m i n a d a l a d e s c o m p o s i c i o n d e E
3
L a s p o s i b i l i d a d e s s o n :
a ) E
3 1
E
3 2
c o n d i m E
3 1
= d i m E
3 2
= 3
b ) E
3 1
E
3 2
E
3 3
c o n d i m E
3 1
= 3 , d i m E
3 2
= 2 y d i m E
3 3
= 1
c ) E
3 1
E
3 2
E
3 3
E
3 4
c o n d i m E
3 1
= 3 , d i m E
3 2
= d i m E
3 3
= d i m E
3 4
= 1
Y l a m a t r i z f r e s t r i n g i d a a E
3
e s
a )
0
B
B
B
B
B
@
; 2 0 0
1 ; 2 0
0 1 ; 2
; 2 0 0
1 ; 2 0
0 1 ; 2
1
C
C
C
C
C
A
b )
0
B
B
B
B
B
@
; 2 0 0
1 ; 2 0
0 1 ; 2
; 2 0
1 ; 2
; 2
1
C
C
C
C
C
A
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1 0 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
c )
0
B
B
B
B
B
@
; 2 0 0
1 ; 2 0
0 1 ; 2
; 2
; 2
; 2
1
C
C
C
C
C
A
l u e g o , l a s p o s i b l e s f o r m a s d e J o r d a n , s o n
a )
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
; 1 0
1 ; 1
4
4
4
; 2
1 ; 2
1 ; 2
; 2
1 ; 2
1 ; 2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
b )
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
; 1
1 ; 1
4
4
4
; 2
1 ; 2
1 ; 2
; 2
1 ; 2
; 2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
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F o r m a r e d u c i d a d e J o r d a n 1 0 3
c )
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
; 1 0
1 ; 1
4
4
4
; 2
1 ; 2
1 ; 2
; 2
; 2
; 2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
3 > E s l a m a t r i z
0
B
B
B
B
B
B
B
@
0 2
1 3
0 2
1 3
2 0
1 2
2
1
C
C
C
C
C
C
C
A
r e d u c i d a d e J o r d a n d e a l g u n e n d o m o r s m o ?
S o l u c i o n :
S i l o f u e s e e l p o l i n o m i o a n u l a d o r s e r a
P ( t ) = ( ; 2 ; 3 t + t
2
) ( t ; 2 )
2
( ; 2 ; 3 t + t
2
) e s e l p o l i n o m i o a n u l a d o r d e
0 2
1 3
p e r o ; 2 ; 3 t + t
2
n o e s p r i m o :
( ; 2 ; 3 t + t
2
) = ( t ;
3 +
p
1 7
2
) ( t ;
3 ;
p
1 7
2
)
l u e g o l a m a t r i z a n t e r i o r n o p u e d e s e r r e d u c i d a d e J o r d a n .
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1 0 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
4 H a l l a r l a f o r m a r e d u c i d a d e J o r d a n d e l e n d o m o r s m o d e R
4
c u y a m a t r i z e n l a
b a s e n a t u r a l e s
A =
0
B
@
0 1 0 0
0 0 2 0
0 0 0 3
0 0 0 0
1
C
A
S o l u c i o n :
H a l l e m o s e l p o l i n o m i o c a r a c t e r s t i c o :
d e t ( A ; t I ) = t
4
( O b v i o y a q u e l a m a t r i z e s t r i a n g u l a r ) .
d i m K e r ( A ; 0 I ) = 4 ; r a n g o A = 4 ; 3 = 1 ( n o d i a g o n a l i z a , p u e s 1 6= 4 )
d i m K e r ( A ; 0 I )
2
= 4 ; r a n g o A
2
= 4 ; r a n g o
0
B
@
0 0 2 0
0 0 0 6
0 0 0 0
0 0 0 0
1
C
A
= 4 ; 2 = 2
d i m K e r ( A ; 0 I )
3
= 4 ; r a n g o A
3
= 4 ; r a n g o
0
B
@
0 0 0 6
0 0 0 0
0 0 0 0
0 0 0 0
1
C
A
= 4 ; 1 = 3
d i m K e r ( A ; 0 I )
4
= 4 ; r a n g o A
4
= 4 ; r a n g o 0 = 4
L u e g o , t e n e m o s
f n
o
d e s u b e s p a c i o s d e d i m 1 g = d i m K e r f = 1
f n
o
d e s u b e s p a c i o s d e d i m 2 g = d i m K e r f
2
; d i m K e r f = 1
f n
o
d e s u b e s p a c i o s d e d i m 3 g = d i m K e r f
3
; d i m K e r f
2
= 1
f n
o
d e s u b e s p a c i o s d e d i m 4 g = d i m K e r f
4
; d i m K e r f
3
= 1
l u e g o h a y u n s o l o s u b e s p a c i o i r r e d u c i b l e d e d i m 4 y l a m a t r i z r e d u c i d a e s
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F o r m a r e d u c i d a d e J o r d a n 1 0 5
0
B
@
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
1
C
A
N o t a : e s t o y a s e p o d a p r e v e e r , p u e s t o q u e a l s e r
d i m K e r ( A ; 0 I ) = d i m K e r f = 1
e l s u b e s p a c i o d e v e c t o r e s p r o p i o s c o r r e s p o n d i e n t e a l v a l o r p r o p i o c e r o e s d e d i m 1 y e n
c a d a s u b e s p a c i o m o n o g e n o h a y u n s u b e s p a c i o d e d i m 1 i n v a r i a n t e , l u e g o s o l o p u e d e
h a b e r u n m o n o g e n o .
5 D a d o e l e n d o m o r s m o d e R
5
c u y a m a t r i z e n l a b a s e n a t u r a l v i e n e d a d a p o r
0
B
B
B
B
B
B
B
B
B
B
B
@
3
2
1
2
;
1
2
1
2
1
2
1
2
5
2
;
1
2
1
2
;
1
2
1
2
1
2
3
2
1
2
;
1
2
1
2
;
1
2
1
2
5
2
;
1
2
0 0 0 1 2
1
C
C
C
C
C
C
C
C
C
C
C
A
H a l l a r :
a ) p o l i n o m i o s c a r a c t e r s t i c o y a n u l a d o r
b ) l o s s u b e s p a c i o s m o n o g e n o s c o r r e s p o n d i e n t e s
c ) u n a b a s e d e e s t o s s u b e s p a c i o s m o n o g e n o s , d i c i e n d o q u e v e c t o r e s s o n
p r o p i o s y e s c r i b i r e n e s t a b a s e l a m a t r i z d e l e n d o m o r s m o .
S o l u c i o n :
a ) P o l i n o m i o c a r a c t e r s t i c o
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1 0 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
Q ( t ) = d e t ( A ; t I ) = ; ( t ; 2 )
5
v e a m o s e l a n u l a d o r :
d i m K e r ( A ; 2 I ) = 5 ; r a n g o ( A ; 2 I ) = 5 ; 3 = 2
( p o r s e r d e d i m e n s i o n d o s h a b r a d o s v e c t o r e s p r o p i o s l i n e a l m e n t e i n d e p e n d i e n t e s , c o n
v a l o r p r o p i o d o s . P o r t a n t o , c o m o e n c a d a s u b e s p a c i o m o n o g e n o h a y u n v e c t o r p r o p i o ,
h a b r a d o s s u b e s p a c i o s m o n o g e n o s )
d i m K e r ( A ; 2 I )
2
= 5 ; r a n g o ( A ; 2 I )
2
) = 5 ; 1 = 4
d i m K e r ( A ; 2 I )
3
= 5 ; r a n g o ( A ; 2 I )
3
= 5
l u e g o ( t ; 2 )
3
a n u l a a t o d o e l e s p a c i o , l u e g o a l p o l i n o m i o a n u l a d o r e s :
P ( t ) = ( t ; 2 )
3
b ) P o r s e r d i m K e r ( A ; 2 I ) = 2 , h a y d o s m o n o g e n o s d e d i m 1
d i m K e r ( A ; 2 I )
2
; d i m K e r ( A ; 2 I ) = 4 ; 2 = 2 , h a y d o s m o n o g e n o s d e d i m 2
d i m K e r ( A ; 2 I )
3
; d i m K e r ( A ; 2 I )
2
= 5 ; 4 = 1 , h a y u n m o n o g e n o d e d i m 3
l u e g o , h a y u n m o n o g e n o d e d i m 3 , y u n m o n o g e n o d e d i m 2
c ) H a l l e m o s u n a b a s e d e l p r i m e r m o n o g e n o f u
1
u
2
u
3
g
u
1
2 K e r ( A ; 2 I )
3
= R
5
, l u e g o u
1
p u e d e s e r c u a l q u i e r v e c t o r t a l q u e ( A ; 2 I )
2
u
1
6=
0 y p u e s t o q u e
K e r ( A ; 2 I )
2
= f ( x y z t k ) =
1
2
x ;
1
2
y +
1
2
z +
1
2
t ;
1
2
k = 0 g
P o d e m o s t o m a r p o r e j e m p l o u
1
= ( 0 0 1 1 0 ) , e n t o n c e s
u
2
= ( A ; 2 I ) u
1
= ( 0 0 0 1 1 )
u
3
= ( A ; 2 I )
2
u
1
= ( A ; 2 I ) u
2
= ( 1 0 0 0 1 )
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F o r m a r e d u c i d a d e J o r d a n 1 0 7
y u
3
e s v e c t o r p r o p i o ( ( A ; 2 I ) u
3
= ( A ; 2 I )
3
u
1
= 0 u
1
= 0 ) .
H a l l e m o s a h o r a u n a b a s e d e l s e g u n d o m o n o g e n o u
4
u
5
:
u
4
2 K e r ( A ; 2 I )
2
y u
4
= 2 K e r ( A ; 2 I )
K e r ( A ; 2 I )
2
= f ( x y z t k ) =
1
2
x ;
1
2
z +
1
2
t ;
1
2
k = 0 g
K e r ( A ; 2 I ) = f ( x y z t k ) = ; x + y ; z + t + k = 0 x + y ; z + t ; k = 0
x ; y + z + t ; k = 0 y = 0 g
O b s e r v a m o s q u e u
2
2 K e r ( A ; 2 I )
2
u
2
= 2 K e r ( A ; 2 I ) , l u e g o t e n e m o s q u e t o m a r l a
p r e c a u c i o n d e e l e g i r u
4
d e f o r m a q u e u
4
( A ; 2 I ) u
4
= u
5
s e a n l i n e a l m e n t e i n d e p e n -
d i e n t e s d e u
2
u
3
S e a p u e s
u
4
= ( 1 1 0 0 0 )
y p o r l o t a n t o
u
5
= ( A ; 2 I ) u
4
= ( 0 1 1 0 0 )
y u
5
e s v e c t o r p r o p i o .
V a y a m o s a d e t e r m i n a r l a m a t r i z d e f e n l a b a s e f u
1
u
2
u
3
u
4
u
5
g :
( A ; 2 I ) u
1
= u
2
) A u
1
; 2 u
1
= u
2
) A u
1
= 2 u
1
+ u
2
( A ; 2 I ) u
2
= u
3
) A u
1
; 2 u
2
= u
3
) A u
2
= 2 u
2
+ u
3
( A ; 2 I ) u
3
= 0 ) A u
3
; 2 u
3
= 0 ) A u
3
= 2 u
3
( A ; 2 I ) u
4
= u
5
) A u
4
; 2 u
4
= u
5
) A u
4
= 2 u
4
+ u
5
( A ; 2 I ) u
5
= 0 ) A u
5
; 2 u
5
= 0 ) A u
5
= 2 u
5
l u e g o , l a m a t r i z e s
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1 0 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
J =
0
B
B
B
@
2
1 2
1 2
2
1 2
1
C
C
C
A
y c l a r a m e n t e J = S
; 1
A S , d o n d e
S =
0
B
B
B
@
0 0 1 1 0
0 0 0 1 1
1 0 0 0 1
1 1 0 0 0
0 1 1 0 0
1
C
C
C
A
6 S e a f 2 E n d ( R
3
) c u y a m a t r i z e n l a b a s e n a t u r a l e s
A
1
=
0
B
B
B
@
1
1
2
1
0 1 0
0 1 1
1
C
C
C
A
y s e a g 2 E n d ( R
3
) c u y a m a t r i z e n u n a c i e r t a b a s e : B = f v
1
v
2
v
3
g e s
A
2
=
0
@
; 1 ; 7 ; 2
2 6 1
; 2 ; 9 ; 2
1
A
> P u e d e n s e r f y g e l m i s m o e n d o m o r s m o ?
S o l u c i o n :
P a r a q u e A
1
y A
2
p u e d a n r e p r e s e n t a r e l m i s m o e n d o m o r s m o h a d e e x i s t i r u n a m a t r i z
S t a l q u e S
; 1
A
1
S = A
2
V e a m o s c o m o p o d e m o s d e t e r m i n a r d i c h a m a t r i z : b u s q u e m o s ( s i e x i s t e n ) l a s f o r m a s
r e d u c i d a s d e J o r d a n d e f y g . S i s o n e l m i s m o e n d o m o r s m o , c o i n c i d i r a n y t e n d r e m o s
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F o r m a r e d u c i d a d e J o r d a n 1 0 9
A
1
= S
; 1
1
J S
1
A
2
= S
; 1
2
J S
2
c o n l o c u a l t e n d r e m o s
S
1
A
1
S
; 1
1
= J
S
2
A
2
S
; 1
2
= J
)
) S
1
A
1
S
; 1
1
= S
2
A
2
S
; 1
2
) ( S
; 1
1
S
2
)
; 1
A
1
( S
; 1
1
S
2
) = A
2
y S = S
; 1
1
S
2
e s l a m a t r i z b u s c a d a y a q u e s e v e r i c a : S
; 1
A
1
S = A
2
E s t u d i e m o s p u e s A
1
d e t ( A
1
; t I ) = ; ( t ; 1 )
3
d i m K e r ( A
1
; I ) = 1
l u e g o h a y u n s o l o m o n o g e n o y J s e r a
J =
0
@
1 0 0
1 1 0
0 1 1
1
A
y l a b a s e d e J o r d a n e s
w
1
2 K e r ( A
1
; I )
3
= R
3
w
1
= 2 K e r ( A
1
; I )
2
p o r e j e m p l o w
1
= ( 0 1 0 )
w
2
= ( A
1
; I ) w
1
= (
1
2
0 1 )
w
3
= ( A
1
; I )
2
w
1
= ( A ; I ) w
2
= ( 1 0 0 )
y S
1
=
0
B
B
B
@
0 1 0
0 0 1
1 0 ;
1
2
1
C
C
C
A
S
; 1
1
=
0
B
B
B
@
0
1
2
1
1 0 0
0 1 0
1
C
C
C
A
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1 1 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
P a s e m o s a e s t u d i a r , a h o r a , A
2
d e t ( A
2
; t I ) = ; ( t ; 1 )
3
d i m K e r ( A
2
; I ) = 1
l u e g o h a y u n s o l o m o n o g e n o y
J =
0
@
1 0 0
1 1 0
0 1 1
1
A
y l a b a s e d e J o r d a n e s
u
1
2 K e r ( A
2
; I )
3
= R
3
u
1
= 2 K e r ( A
2
; I )
2
p o r e j e m p l o u
1
= ( 8 ; 5 1 0 )
u
2
= ( A
2
; I ) u
1
= ( 1 ; 1 1 )
u
3
= ( A
2
; I )
2
u
1
= ( A
2
; I ) u
2
= ( ; 3 2 ; 4 )
y S
2
=
0
@
2 1 ; 1
0 2 1
5 2 ; 3
1
A
S
; 1
2
=
0
@
8 ; 1 ; 3
; 5 1 2
1 0 ; 1 ; 4
1
A
y p o r l o t a n t o :
S = S
; 1
1
S
2
=
0
B
B
B
@
5 3 ;
5
2
2 1 ; 1
0 2 1
1
C
C
C
A
7 S e a f = ( D + I ) : P
2
( R ) ; ! P
2
( R ) d o n d e P
2
( R ) e s e l e s p a c i o d e p o l i n o m i o s d e
g r a d o m e n o r o i g u a l q u e d o s a c o e c i e n t e s r e a l e s y D e s l a a p l i c a c i o n d e r i v a d a
a ) d e t e r m i n a r l a f o r m a r e d u c i d a d e J o r d a n a s c o m o l a b a s e p a r a l a c u a l l a m a t r i z
a d o p t a d i c h a f o r m a
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F o r m a r e d u c i d a d e J o r d a n 1 1 1
b ) p r o b a r q u e f
; 1
e s u n p o l i n o m i o e n f y u t i l i z a r d i c h o r e s u l t a d o p a r a d e t e r m i n a r
l a m a t r i z d e f
; 1
e n l a b a s e n a t u r a l f x
2
x 1 g
S o l u c i o n :
a ) E n l a b a s e f x
2
x 1 g , l a m a t r i z d e f a d o p t a l a f o r m a
A =
0
@
1 0 0
2 1 0
0 1 1
1
A
d e t ( A ; t I ) = ; ( t ; 1 )
3
d i m K e r ( A ; I ) = 3 ; r a n g o ( A ; I ) = 3 ; 2 = 1
l u e g o h a y u n s o l o m o n o g e n o y l a m a t r i z r e d u c i d a d e J o r d a n e s
J =
0
@
1 0 0
1 1 0
0 1 1
1
A
B u s q u e m o s l a b a s e d e J o r d a n :
v
1
2 K e r ( A ; I )
3
= R
3
v
1
= 2 K e r ( A ; I )
2
= f ( x y z ) = x = 0 g
v
1
= ( 1 0 0 )
v
2
= ( A ; I ) v
1
= ( 0 2 0 )
v
3
= ( A ; I )
2
v
1
= ( A ; I ) v
2
= ( 0 0 2 )
l u e g o l a b a s e e s f ( 1 0 0 ) ( 0 2 0 ) ( 0 0 2 ) g
b ) E l p o l i n o m i o a n u l a d o r d e f e s ( t ; 1 )
3
, l u e g o
( f ; I )
3
= 0 , f
3
; 3 f
2
+ 3 f ; I = 0
l u e g o
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1 1 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
I = f
3
; 3 f
2
+ 3 f = f ( f
2
; 3 f + 3 I ) = ( f
2
; 3 f + 3 I ) f
p o r l o q u e
f
; 1
= f
2
; 3 f + 3 I
Y l a m a t r i z A
; 1
e s :
A
; 1
=
0
@
1 0 0
2 1 0
0 1 1
1
A
2
; 3
0
@
1 0 0
2 1 0
0 1 1
1
A
+ 3
0
@
1
1
1
1
A
=
=
0
@
1 0 0
; 2 1 0
2 ; 1 1
1
A
8 H a l l a r l a f o r m a n o r m a l d e J o r d a n d e l e n d o m o r s m o d e R
4
c u y a m a t r i z e s
A =
0
B
@
3 1 0 0
; 4 ; 1 0 0
7 1 2 1
; 1 7 ; 6 ; 1 0
1
C
A
H a l l a n d o l a b a s e d e R
4
e n l a c u a l l a m a t r i z d e l e n d o m o r s m o a d o p t a d i c h a
f o r m a n o r m a l .
S o l u c i o n :
C a l c u l e m o s l o s p o l i n o m i o s c a r a c t e r s t i c o y a n u l a d o r d e A
d e t ( A ; t I ) = d e t ( (
3 1
; 4 ; 1
) ; t I
2
) d e t ( (
2 1
; 1 0
) ; t I
2
) =
= ( t ; 1 )
4
d i m K e r ( A ; t I ) = 4 ; r a n g o ( A ; I ) = 4 ; 2 = 2
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F o r m a r e d u c i d a d e J o r d a n 1 1 3
l u e g o h a y d o s m o n o g e n o s
d i m K e r ( A ; I )
2
= 4 ; r a n g o ( A ; I )
2
= 4 ; 0 = 4 = d i m R
4
l u e g o s o n d o s m o n o g e n o s d e d i m 2 y l a f o r m a d e J o r d a n e s
J =
0
B
@
1 0 0 0
1 1 0 0
0 0 1 0
0 0 1 1
1
C
A
Y R
4
= E
1
E
2
c o n d i m E
i
= 2 p a r a i = 1 2
B u s q u e m o s l a b a s e d e J o r d a n :
b a s e d e E
1
:
v
1
2 K e r ( A ; I )
2
v
1
= 2 K e r ( A ; I )
v
1
= ( 1 0 0 0 )
v
2
= ( A ; I ) v
1
= ( 2 ; 4 7 ; 1 7 )
b a s e d e E
2
:
v
3
2 K e r ( A ; I )
2
v
3
= 2 K e r ( A ; I )
v
3
= ( 0 1 0 0 )
v
4
= ( A ; I ) v
3
= ( 1 ; 2 1 ; 6 )
h a y q u e t o m a r l a p r e c a u c i o n d e q u e v
3
v
4
s e a n l i n e a l m e n t e i n d e p e n d i e n t e s d e v
1
v
2
9 D e t e r m i n a r l a f o r m a r e d u c i d a d e J o r d a n d e l e n d o m o r s m o d e R
3
c u y a m a t r i z e n
l a b a s e n a t u r a l e s
A =
0
@
1 a a
; 1 1 ; 1
1 0 2
1
A
c o n a 2 R
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1 1 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S o l u c i o n :
B u s q u e m o s e l p o l i n o m i o c a r a c t e r s t i c o :
d e t ( A ; t I ) = ; ( t ; 1 )
2
( t ; 2 )
d i m K e r ( A ; I ) =
(
2 a = 0
1 a 6= 0
P a r a a = 0 f d i a g o n a l i z a , y D e s
D =
0
@
1 0 0
0 1 0
0 0 2
1
A
p a r a a 6= 0 e l v a l o r p r o p i o 1 n o s d a u n u n i c o s u b e s p a c i o m o n o g e n o , y J e s
J =
0
@
1 0 0
1 1 0
0 0 2
1
A
B u s q u e m o s l a b a s e d e J o r d a n : d i s t i n g u i r e m o s d o s c a s o s
1 ) a = 0
v
1
v
2
2 K e r ( A ; I ) = f ( x y z ) = x + z = 0 g
e l e g i m o s v
1
= ( 1 0 ; 1 ) v
2
= ( 0 1 0 )
v
3
2 K e r ( A ; 2 I ) = f ( x y z ) = x = 0 y + z = 0 g
e l e g i m o s v
3
= ( 0 1 ; 1 )
2 ) a 6= 0
v
1
2 K e r ( A ; I )
2
= f ( x y z ) = x + a y + ( a + 1 ) z = 0 g
v
1
= 2 K e r ( A ; I ) = f ( x y z ) = a y + a z = 0 x + z = 0 g
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F o r m a r e d u c i d a d e J o r d a n 1 1 5
e l e g i m o s v
1
= ( a ; 1 0 )
v
2
= ( A ; I ) v
1
= ( ; a ; a a )
v
3
2 K e r ( A ; 2 I )
v
3
= ( 0 1 ; 1 )
1 0 S e a A 2 M
n
( R ) y s e a H e l R - e s p a c i o v e c t o r i a l g e n e r a d o p o r l a s m a t r i c e s
f I A A
2
A
n ; 1
g
a ) D e m o s t r a r q u e s i B 2 H y B e s i n v e r s i b l e , e n t o n c e s B
; 1
2 H
b ) S i d e t A = 0 , p r o b a r q u e e x i s t e B 2 H B 6= 0 t a l q u e A B = B A = 0
S o l u c i o n :
a ) P o r e l t e o r e m a d e C a y l e y - H a m i l t o n , s a b e m o s q u e e l p o l i n o m i o c a r a c t e r s t i c o
n
+
1
n ; 1
+ +
n
a n u l a a l a m a t r i z :
A
n
+
1
A
n ; 1
+ +
n
I = 0
p o r l o q u e A
n
=
P
n
i = 1
;
i
A
n ; i
2 H , c o n l o c u a l A
m
2 H 8 m n , y t i e n e s e n t i d o
l a a p l i c a c i o n :
f : H ; ! H
C ; ! B C
f e s l i n e a l , p u e s
f ( C
1
+ C
2
) = B ( C
1
+ C
2
) = B C
1
+ B C
2
= f ( C
1
) + f ( C
2
)
f ( C ) = B ( C ) = B C = f ( C )
f e s i n y e c t i v a , p u e s s i B C
1
= B C
2
, a l s e r B i n v e r s i b l e , t e n e m o s B
; 1
( B C
1
) =
B
; 1
( B C
2
) y p o r t a n t o , C
1
= C
2
, y p u e s t o q u e H e s d e d i m e n s i o n n i t a f e s
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1 1 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
b i y e c t i v a , l u e g o I 2 H t i e n e a n t i i m a g e n p o r l a a p l i c a c i o n f e s d e c i r , e x i s t e C 2 H
t a l q u e f ( C ) = B C = I , l u e g o C = B
; 1
2 H
N o t a : p u e s t o q u e l a s m a t r i c e s s o n d e o r d e n n i t o d e B C = I , d e d u c i m o s C = B
; 1
S i f u e r a n d e o r d e n i n n i t o , p o d r a s e r q u e B C = I , p e r o C B 6= I
b ) S u p o n g a m o s A 6= 0 s e a p ( ) e l p o l i n o m i o a n u l a d o r d e A t e n e m o s q u e p ( A ) =
P
r
i = 0
i
A
i
= 0 y p u e s t o q u e d e t A = 0 e s
0
= 0 ( y a q u e e l p o l i n o m i o a n u l a d o r
d i v i d e a l c a r a c t e r s t i c o y t i e n e s u s m i s m a s r a c e s ) , l u e g o
1
A + +
r
A
r
= 0
y s e a p u e s B =
1
I + +
r
A
r ; 1
B e s d i s t i n t a d e c e r o , y a q u e s i B = 0 e l p o l i n o m i o a n u l a d o r d e A s e r a
1
+ +
r
x
r ; 1
S i A = 0 , e n t o n c e s 8 B 2 H , t e n e m o s A B = B A = 0
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A n a l i s i s m a t r i c i a l 1 1 7
C a p t u l o 8 A n a l i s i s m a t r i c i a l
1 D a d a l a m a t r i z
A =
0
@
3 2 4
2 0 2
4 2 3
1
A
a ) C a l c u l a r e
A
e
t A
b ) U t i l i z a r d i c h o r e s u l t a d o p a r a r e s o l v e r e l s i g u i e n t e s i s t e m a d e e c u a c i o n e s d i f e r e n -
c i a l e s :
8
>
<
>
:
x
0
= 3 x + 2 y + 4 z
y
0
= 2 x + 2 z
z
0
= 4 x + 2 y + 3 z
s a b i e n d o q u e p a r a t = 0 x = 1 y = 2 z = 3
S o l u c i o n :
a ) L a e x p o n e n c i a l d e u n a m a t r i z v i e n e d e n i d a p o r :
e
A
= l i m
p ! 1
( I + A +
1
2 !
A
2
+
1
p !
A
p
)
P u e s t o q u e e x i s t e S t a l q u e A = S D S
; 1
, c o n D m a t r i z d i a g o n a l , t e n e m o s q u e :
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1 1 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
e
A
= l i m
p ! 1
( S S
; 1
+ S D S
; 1
+
1
2 !
S D
2
S
; 1
+ +
1
p !
S D
p
S
; 1
) =
= S l i m
p ! 1
( I + D +
1
2 !
D
2
+
1
p !
D
p
) S
; 1
= S e
D
S
; 1
v e a m o s q u e e n e f e c t o e x i s t e n l a s m a t r i c e s S y D
d e t ( A ; I ) = ; ( + 1 )
2
( ; 8 )
d i m k e r ( A + I ) = 2
l u e g o
D =
0
@
; 1 0 0
0 ; 1 0
0 0 8
1
A
d e t e r m i n e m o s S
f v
1
v
2
g b a s e d e k e r ( A + I )
0
@
4 2 4
2 1 2
4 2 4
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
) 2 x + y + 2 z = 0 )
v
1
= ( 1 0 ; 1 )
v
2
= ( 0 2 ; 1 )
9
=
v
3
2 k e r ( A ; 8 I )
0
@
; 5 2 4
2 ; 8 2
4 2 ; 5
1
A
0
@
x
y
z
1
A
=
0
@
0
0
0
1
A
)
; 5 x + 2 y + 4 z = 0
2 x ; 8 y + 2 z = 0
9
=
) v
3
= ( 2 1 2 )
d e d o n d e
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A n a l i s i s m a t r i c i a l 1 1 9
S =
0
@
1 0 2
0 2 1
; 1 ; 1 2
1
A
y S
; 1
=
1
9
0
@
5 ; 2 ; 4
; 1 4 ; 1
2 1 2
1
A
p o r l o t a n t o
e
D
= l i m
p ! 1
0
@
0
@
1
1
1
1
A
+
0
@
; 1
; 1
8
1
A
+
1
2 !
0
@
( ; 1 )
2
( ; 1 )
2
( 8 )
2
1
A
+
1
p !
0
@
( ; 1 )
p
( ; 1 )
p
( 8 )
p
1
A
1
A
=
0
@
e
; 1
e
; 1
e
8
1
A
y
e
A
= S e
D
S
; 1
=
1
9
0
@
5 e
; 1
+ 4 e
8
; 2 e
; 1
+ 2 e
8
; 4 e
; 1
+ 4 e
8
; 2 e
; 1
+ 2 e
8
8 e
; 1
+ e
8
; 2 e
; 1
+ 2 e
8
; 4 e
; 1
+ 4 e
8
; 2 e
; 1
+ 2 e
8
5 e
; 1
+ 4 e
8
1
A
e
t A
= l i m
p ! 1
( I + t A +
1
2 !
t
2
A
2
+ +
1
p !
t
p
A
p
) = S e
t D
S
; 1
y
e
t D
=
0
@
e
; t
0 0
0 e
; t
0
0 0 e
8 t
1
A
p o r l o q u e
e
t A
=
1
9
0
@
5 e
; t
+ 4 e
8 t
; 2 e
; t
+ 2 e
8 t
; 4 e
; t
+ 4 e
8 t
; 2 e
; t
+ 2 e
8 t
8 e
; t
+ e
8 t
; 2 e
; t
+ 2 e
8 t
; 4 e
; t
+ 4 e
8 t
; 2 e
t
+ 2 e
8 t
5 e
; t
+ 4 e
8 t
1
A
b ) P a s e m o s a l a r e s o l u c i o n d e l s i s t e m a d e e c u a c i o n e s d i f e r e n c i a l e s
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1 2 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
X ( t ) = e
t A
X ( 0 ) c o n X ( 0 ) =
0
@
1
2
3
1
A
X ( t ) =
0
@
x ( t )
y ( t )
z ( t )
1
A
p o r l o q u e
X ( t ) =
1
9
0
@
; 1 1 e
; t
+ 2 0 e
8 t
8 e
; t
+ 1 0 e
8 t
7 e
; t
+ 2 0 e
8 t
1
A
2 R e s o l v e r e l s i s t e m a d e e c u a c i o n e s d i f e r e n c i a l e s
d x
d t
= x + 2 y ; 4 z
d y
d t
= ; y + 6 z
d z
d t
= ; y + 4 z
9
>
>
>
>
>
=
>
>
>
>
>
S o l u c i o n :
E l s i s t e m a p u e d e e x p r e s a r s e m a t r i c i a l m e n t e
0
@
d x
d t
d y
d t
d z
d t
1
A
=
0
@
1 2 ; 4
0 ; 1 6
0 ; 1 4
1
A
0
@
x
y
z
1
A
e s d e c i r ,
d X
d t
= A X
I n t e n t a r e m o s e f e c t u a r u n c a m b i o d e b a s e d e m o d o q u e l a n u e v a m a t r i z J = S
; 1
A S s e a
l o m a s s e n c i l l a p o s i b l e . A s , s i X = S Z , t e n e m o s
d X
d t
= S
d Z
d t
y l a e c u a c i o n q u e d a
S
d Z
d t
= S J S
; 1
S Z , e s d e c i r
d Z
d t
= J Z
B u s q u e m o s l a f o r m a r e d u c i d a d e J o r d a n d e A
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A n a l i s i s m a t r i c i a l 1 2 1
d e t ( A ; I ) = ; ( ; 1 )
2
( ; 2 )
d i m k e r ( A ; I ) = 1
l u e g o n o d i a g o n a l i z a y
J =
0
@
1 0 0
1 1 0
0 0 2
1
A
L a b a s e d e J o r d a n e s
v
1
2 k e r ( A ; I )
2
v
1
= 2 k e r ( A ; I ) s e a p u e s v
1
= ( 1 3 1 )
v
2
= ( A ; I ) ( v
1
) = ( 2 0 0 )
v
3
2 k e r ( A ; 2 I ) s e a p u e s v
3
= ( 0 2 1 )
y l a m a t r i z S e s
S =
0
@
1 2 0
3 0 2
1 0 1
1
A
E l s i s t e m a q u e d a
0
@
d z
1
d t
d z
2
d t
d z
3
d t
1
A
=
0
@
1 0 0
1 1 0
0 0 2
1
A
0
@
z
1
z
2
z
3
1
A
c u y a s o l u c i o n e s :
z
1
= C
1
e
t
z
2
= ( C
1
t + C
2
) e
t
z
3
= C
3
e
2 t
9
>
=
>
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1 2 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
q u e v o l v i e n d o a l a b a s e n a t u r a l
0
@
x
y
z
1
A
=
0
@
1 2 0
3 0 2
1 0 1
1
A
0
@
C
1
e
t
( C
1
t + C
2
) e
t
C
3
e
2 t
1
A
d e d o n d e
x = ( 2 C
1
t + C
1
+ 2 C
2
) e
t
y = 3 C
1
e
t
+ 2 C
3
e
2 t
z = C
1
e
t
+ C
3
e
2 t
9
>
=
>
3 S e a f u n e n d o m o r s m o d e l R - e s p a c i o v e c t o r i a l R
4
t a l q u e s u m a t r i z e n l a b a s e
n a t u r a l e s :
A =
0
B
@
; 1 0 0 0
1 2 9 ; 4 ; 4
3 0 2 5 ; 1 1 ; 1 3
0 0 0 1
1
C
A
a ) O b t e n e r l a f o r m a r e d u c i d a d e J o r d a n d e f y l a b a s e d e J o r d a n c o r r e s p o n d i e n t e .
b ) C a l c u l a r e
3 A
S o l u c i o n :
a ) D e t e r m i n e m o s l a f o r m a r e d u c i d a d e A
d e t ( A ; I ) = ( + 1 ) 3 ( ; 1 )
d i m k e r ( A + I ) = 2
l u e g o n o d i a g o n a l i z a , y e l v a l o r p r o p i o ; 1 n o s p r o p o r c i o n a d o s m o n o g e n o s y l a m a t r i z
d e J o r d a n e s
J =
0
B
@
; 1 0 0 0
1 ; 1 0 0
0 0 ; 1 0
0 0 0 1
1
C
A
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A n a l i s i s m a t r i c i a l 1 2 3
B u s q u e m o s l a b a s e d e J o r d a n
v
1
2 k e r ( A + I )
2
v
1
= 2 k e r ( A + I ) v
1
= ( 1 0 0 0 )
v
2
= ( A + I ) v
1
= ( 0 1 2 3 0 0 )
v
3
2 k e r ( A + I ) i n d e p e n d i e n t e c o n v
2
s e a v
3
= ( 1 0 3 0 )
v
4
2 k e r ( A + I ) v
4
= ( 0 1 1 1 )
y l a m a t r i z c a m b i o d e b a s e e s
S =
0
B
@
1 0 1 0
0 1 2 0 1
0 3 0 3 1
0 0 0 1
1
C
A
y S
; 1
=
1
1 2
0
B
@
1 2 1 0 ; 4 ; 6
0 1 0 ; 1
0 ; 1 0 4 6
0 0 0 1 2
1
C
A
b ) e
3 A
= e
3 S J S
1
= S e
3 J
S
; 1
e
3 J
=
0
B
@
e
; 3
0 0 0
3 e
; 3
e
; 3
0 0
0 0 e
; 3
0
0 0 0 e
3
1
C
A
l u e g o
e
3 A
=
0
B
@
e
; 3
0 0 0
3 6 e
; 3
3 1 e
; 3
; 1 2 e
; 3
; 1 9 e
; 3
+ e
3
9 0 e
; 3
7 5 e
; 3
; 2 9 e
; 3
; 4 6 e
; 3
+ e
3
0 0 0 e
3
1
C
A
4 D e t e r m i n a r l a s f u n c i o n e s r e a l e s d e u n a v a r i a b l e x ( t ) , y ( t ) , z ( t ) , u ( t ) t a l e s q u e
v e r i c a n e l s i g u i e n t e s i s t e m a d e e c u a c i o n e s d i f e r e n c i a l e s l i n e a l e s
x
0
= x ; z + u
y
0
= y + z
z
0
= z
u
0
= u
9
>
>
>
=
>
>
>
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1 2 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
y l a s c o n d i c i o n e s i n i c i a l e s x ( 0 ) = 1 y ( 0 ) = 0 z ( 0 ) = 1 u ( 0 ) = 2 .
S o l u c i o n :
E s c r i b i e n d o e l s i s t e m a d a d o e n f o r m a m a t r i c i a l A X = X
0
0
B
@
1 0 ; 1 1
0 1 1 0
0 0 1 0
0 0 0 1
1
C
A
0
B
@
x
y
z
u
1
C
A
=
0
B
@
x
0
y
0
z
0
u
0
1
C
A
B u s q u e m o s l a f o r m a r e d u c i d a d e J o r d a n d e l a m a t r i z A p a r a s i m p l i c a r e l p r o b l e m a
d e t ( A ; I ) = ( ; 1 )
4
d i m k e r ( A ; I ) = 2
l u e g o h a y d o s m o n o g e n o s
d i m k e r ( A ; I )
2
= 4
l u e g o a m b o s m o n o g e n o s s o n d e d i m e n s i o n d o s , p o r l o q u e l a m a t r i z d e J o r d a n a d o p t a
l a f o r m a
J =
0
B
@
1 0 0 0
1 1 0 0
0 0 1 0
0 0 1 1
1
C
A
B u s q u e m o s l a b a s e d e J o r d a n
e
1
e
3
2 k e r ( A ; I )
2
e
1
e
3
= 2 k e r ( A ; I )
e
2
= ( A ; I ) e
1
e
4
= ( A ; I ) e
3
d e m a n e r a q u e e
1
e
2
e
3
e
4
s e a n i n d e p e n d i e n t e s .
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A n a l i s i s m a t r i c i a l 1 2 5
S e a p u e s
e
1
= ( 0 0 1 0 ) ) e
2
= ( ; 1 1 0 0 )
e
3
= ( 0 0 0 1 ) ) e
4
= ( 1 0 0 0 )
l u e g o
S =
0
B
@
0 ; 1 0 1
0 1 0 0
1 0 0 0
0 0 1 0
1
C
A
y S
; 1
=
0
B
@
0 0 1 0
0 1 0 0
0 0 0 1
1 1 0 0
1
C
A
e
t A
= e
t S J S
1
= S e
t J
S
; 1
e
t J
= e
t
0
B
@
1 0 0 0
t 1 0 0
0 0 1 0
0 0 t 1
1
C
A
=
0
B
@
e
t
0 0 0
t e
t
e
t
0 0
0 0 e
t
0
0 0 t e
t
e
t
1
C
A
p o r l o t a n t o
e
t A
=
0
B
@
e
t
0 ; t e
t
t e
t
0 e
t
t e
t
0
0 0 e
t
0
0 0 0 e
t
1
C
A
y l a s o l u c i o n d e l s i s t e m a e s :
0
B
@
x
y
z
u
1
C
A
= e
t A
0
B
@
x ( 0 )
y ( 0 )
z ( 0 )
u ( 0 )
1
C
A
=
0
B
@
e
t
t e
t
e
t
e
t
1
C
A
5 D a d a l a m a t r i z
A =
0
1
6
1
1
6
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1 2 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
H a l l a r :
I + A + A
2
+ + A
n
+ =
1
X
i = 0
A
i
S o l u c i o n :
B u s q u e m o s , p a r a o b t e n e r d e f o r m a s e n c i l l a A
n
, l a f o r m a r e d u c i d a d e J o r d a n d e l a
m a t r i z A
d e t ( A ; I ) = ( ;
1
2
) ( +
1
3
)
l u e g o A d i a g o n a l i z a
D =
1
2
0
0 ;
1
3
y l a m a t r i z c a m b i o d e b a s e e s :
v
1
2 k e r ( A ;
1
2
I ) v
1
= ( 1 3 )
v
2
2 k e r ( A +
1
3
I ) v
2
= ( ; 1 2 )
S =
1 ; 1
3 2
y S
; 1
=
1
5
2 1
; 3 1
A
n
= S D
n
S
; 1
= S
(
1
2
)
n
( ;
1
3
)
n
S
; 1
=
=
0
@
2
5
(
1
2
)
n
+
3
5
(
1
3
)
n
1
5
(
1
2
)
n
;
1
5
(
1
3
)
n
6
5
(
1
2
)
n
;
6
5
( ;
1
3
)
n
3
5
(
1
2
)
n
+
2
5
( ;
1
3
)
n
1
A
1
X
n = 0
A
n
=
0
@
2
5
P
1
n = 0
(
1
2
)
n
+
3
5
P
1
n = 0
(
1
3
)
n
1
5
P
1
n = 0
(
1
2
)
n
;
1
5
P
1
n = 0
(
1
3
)
n
6
5
P
1
n = 0
( ;
1
2
)
n
;
6
5
P
1
n = 0
( ;
1
3
)
n
3
5
P
1
n = 0
(
1
2
)
n
+
2
5
P
1
n = 0
( ;
1
3
)
n
1
A
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A n a l i s i s m a t r i c i a l 1 2 7
P
1
n = 0
(
1
2
)
n
= 2 ( e s l a s u m a d e l o s t e r m i n o s d e u n a p r o g r e s i o n g e o m e t r i c a d e p r i m e r
t e r m i n o 1 y r a z o n
1
2
< 1 )
P
1
n = 0
( ;
1
3
)
n
=
3
4
( e s l a s u m a d e l o s t e r m i n o s d e u n a p r o g r e s i o n g e o m e t r i c a d e
p r i m e r t e r m i n o 1 y r a z o n ;
1
3
;
1
3
< 1 )
P o r l o q u e :
2
5
1
X
n = 0
(
1
2
)
n
+
3
5
1
X
n = 0
( ;
1
3
)
n
=
5
4
1
5
1
X
n = 0
(
1
2
)
n
;
1
5
1
X
n = 0
( ;
1
3
)
n
=
1
4
6
5
1
X
n = 0
(
1
2
)
n
;
6
5
1
X
n = 0
( ;
1
3
)
n
=
3
2
3
5
1
X
n = 0
(
1
2
)
n
+
2
5
1
X
n = 0
( ;
1
3
)
n
=
3
2
y
1
X
n = 0
A
n
=
1
4
5 4
6 6
6 S e a
A =
0
@
0 1 1
1 0 1
1 1 0
1
A
C a l c u l a r s e n A
S o l u c i o n :
P o r d e n i c i o n :
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1 2 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
s e n A =
1
X
n = 0
( ; 1 )
n
A
2 n + 1
( 2 n + 1 ) !
D e t e r m i n e m o s l a f o r m a r e d u c i d a d e J o r d a n d e A
d e t ( A ; I ) = ; ( + I )
2
+ ( ; 2 )
d i m k e r ( A + I ) = 2
l u e g o A d i a g o n a l i z a .
L a m a t r i z c a m b i o d e b a s e e s
v
1
v
2
2 k e r ( A + I ) i n d e p e n d i e n t e s v
3
2 k e r ( A ; I ) s e a n p u e s
v
1
= (
p
2
2
;
p
2
2
0 )
v
2
= (
p
6
6
p
6
6
;
2
p
6
2
)
v
3
= (
p
3
3
p
3
3
p
3
3
)
l u e g o
S =
0
B
B
B
B
@
p
2
2
p
6
6
p
3
3
;
p
2
2
p
6
6
p
3
3
0 ;
2
p
6
6
p
3
3
1
C
C
C
C
A
y S
; 1
=
0
B
B
B
B
@
p
2
2
;
p
2
2
0
p
6
6
p
6
6
;
2
p
6
6
p
3
3
p
3
3
p
3
3
1
C
C
C
C
A
y p o r l o t a n t o
s e n A =
1
X
n = 0
( ; 1 )
n
S D
2 n + 1
S
; 1
( 2 n + 1 ) !
= S (
1
X
n = 0
( ; 1 )
n
D
2 n + 1
( 2 n + 1 ) !
) S
; 1
=
= S
0
B
B
B
B
@
P
1
n = 0
( ; 1 )
n
( 2 n + 1 ) !
( ; 1 )
2 n + 1
0 0
0
P
1
n = 0
( ; 1 )
n
( 2 n + 1 ) !
( ; 1 )
2 n + 1
0
0 0
P
1
n = 0
( ; 1 )
n
( 2 n + 1 ) !
( 2 )
2 n + 1
1
C
C
C
C
A
S
; 1
=
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A n a l i s i s m a t r i c i a l 1 2 9
= S
0
B
B
B
@
s e n ( ; 1 ) 0 0
0 s e n ( ; 1 ) 0
0 0 s e n ( 2 )
1
C
C
C
A
S
; 1
=
=
0
B
B
B
@
2
3
s e n ( ; 1 ) ;
1
3
s e n ( ; 1 ) +
1
3
s e n ( 2 ) ;
1
3
s e n ( ; 1 ) +
1
3
s e n ( 2 )
;
1
3
s e n ( ; 1 ) +
1
3
s e n ( 2 )
2
3
s e n ( ; 1 ) +
1
3
s e n ( 2 ) ;
1
3
s e n ( ; 1 ) +
1
3
s e n ( 2 )
;
1
3
s e n ( ; 1 ) +
1
3
s e n ( 2 ) ;
1
3
s e n ( ; 1 ) +
1
3
s e n ( 2 )
2
3
s e n ( ; 1 ) +
1
3
s e n ( 2 )
1
C
C
C
A
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G r u p o s 1 3 1
A p e n d i c e I G r u p o s
1 C o n s i d e r e m o s e l s u b c o n j u n t o G L
2
( R ) d e M
2
( R ) d e n i d o p o r
G L
2
( R ) =
a b
c d
a d ; b c 6= 0
a ) P r o b a r q u e ( G L
2
( R ) ) e s u n g r u p o n o c o n m u t a t i v o , ( e s e l p r o d u c t o h a b i t u a l
e n t r e m a t r i c e s ) .
b ) C o n s i d e r e m o s e l s u b c o n j u n t o S L
2
( R ) d e M
2
( R ) d e n i d o p o r
S L
2
( R ) =
a b
c d
a d ; b c = 1
P r o b a r q u e ( S L
2
( R ) ) e s u n s u b g r u p o d e l g r u p o ( G L
2
( R ) )
S o l u c i o n :
a ) P r i m e r o v e a m o s q u e l a o p e r a c i o n e s t a b i e n d e n i d a , e s d e c i r d a d a s
a b
c d
a
1
b
1
c
1
d
1
2
G l
2
( R ) e n t o n c e s
a b
c d
a
1
b
1
c
1
d
1
=
a a
1
+ b c
1
a b
1
+ b d
1
c a
1
+ d c
1
c b
1
+ d d
1
=
a
2
b
2
c
2
d
2
2 G l
2
( R )
p a r a e l l o b a s t a c a l c u l a r a
2
d
2
; b
2
c
2
a
2
d
2
; b
2
c
2
= ( a a
1
+ b c
1
) ( c b
1
+ d d
1
) ; ( c a
1
+ b c
1
) ( a b
1
+ b d
1
) =
= ( a d ; b c ) ( a
1
d
1
; b
1
c
1
) 6= 0
V e a m o s q u e s e v e r i c a n l a s p r o p i e d a d e s d e g r u p o y q u e f a l l a l a c o n m u t a t i v i d a d
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1 3 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
A s o c i a t i v i d a d
a b
c d
a
1
b
1
c
1
d
1
a
1
b
2
c
2
d
2
=
a a
1
+ b c
1
a b
1
+ b d
1
c a
1
+ d c
1
c b
1
+ d d
1
a
2
b
2
c
2
d
2
=
( a a
1
+ b c
1
) a
2
+ ( a b
1
+ b d
1
) c
2
( a a
1
+ b c
1
) b
2
+ ( a b
1
+ b d
1
) d
2
( c a
1
+ d c
1
) a
2
+ ( c b
1
+ d d
1
) c
2
( c a
1
+ d c
1
) b
2
+ ( c b
1
+ d d
1
) d
2
=
a ( a
1
a
2
+ b
1
c
2
) + b ( c
1
a
2
+ d
1
c
2
) a ( a
1
b
2
+ b
1
d
2
) + d ( c
1
b
2
+ d
1
d
2
)
c ( a
1
a
2
+ b
1
c
2
) + d ( c
1
a
2
+ d
1
c
2
) c ( a
1
b
2
+ b
1
d
2
) + d ( c
1
b
2
+ d
1
d
2
)
=
a b
c d
a
1
a
2
+ b
1
c
2
a
1
b
2
+ b
1
d
2
c
1
a
2
+ d
1
c
2
c
1
b
2
+ d
1
d
2
=
a b
c d
a
1
b
1
c
1
d
1
a
2
b
2
c
2
d
2
E x i s t e n c i a d e e l e m e n t o n e u t r o
8
a b
c d
2 G L
2
( R ) 9
1 0
0 1
t a l q u e
a b
c d
1 0
0 1
=
1 0
0 1
a b
c d
=
a b
c d
y e l e l e m e n t o n e u t r o e s u n i c o : s u p o n g a m o s q u e e x i s t e u n e l e m e n t o
x y
z t
t a l q u e
8
a b
c d
2 G L ( R ) s e t i e n e
a b
c d
x y
z t
=
x y
z t
a b
c d
=
a b
c d
e n t o n c e s
p o r s e r
x y
z t
n e u t r o )
x y
z t
1 0
0 1
=
1 0
0 1
p o r s e r
1 0
0 1
n e u t r o )
x y
z t
1 0
0 1
=
x y
z t
9
>
>
=
>
>
)
x y
z t
=
1 0
0 1
E x i s t e n c i a d e e l e m e n t o s i m e t r i c o
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G r u p o s 1 3 3
8
a b
c d
2 G L
2
( R ) a d ; c b 6= 0 l u e g o 1 = a d ; b c 2 R
S e a
d = a d ; b c ; b = a d ; b c
; c = a d ; b c a = a d ; b c
2 G L
2
( R ) y e s t a l q u e :
d = a d ; b c ; b = a d ; b c
; c = a d ; b c a = a d ; b c
a b
c d
=
a b
c d
d = a d ; b c ; b = a d ; b c
; c = a d ; b c a = a d ; b c
=
1 0
0 1
C l a r a m e n t e p a r a c a d a
a b
c d
2 G L
2
( R ) , e l e l e m e n t o s i m e t r i c o e s u n i c o . ( < C o m p r o -
b a r l o ! )
L u e g o G L
2
( R ) t i e n e e s t r u c t u r a d e g r u p o , v e a m o s q u e n o e s a b e l i a n o .
S e a n
1 1
0 1
1 0
1 1
2 G L
2
( R )
1 1
0 1
1 0
1 1
=
2 1
1 1
1 0
1 1
1 1
0 1
=
1 1
1 2
b ) S e a n A B 2 S l
2
( R ) G L
2
( R ) c o n s i d e r e m o s B
; 1
A 2 G L
2
( R ) y v e a m o s s i
p e r t e n e c e a S L
2
( R )
d e t ( B
; 1
A ) = d e t B
; 1
d e t A = 1 = d e t b d e t A = 1 = 1 1 = 1
2 S e a G u n g r u p o t a l q u e p a r a c a d a x 2 G x
2
= e , s i e n d o e e l e l e m e n t o n e u t r o d e l
g r u p o G
P r o b a r q u e G e s u n g r u p o c o n m u t a t i v o .
S o l u c i o n :
D e x
2
= e s e t i e n e x = x
; 1
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1 3 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
P a r a t o d o p a r d e e l e m e n t o s x y 2 G s e t i e n e x y 2 G l u e g o
( x y )
2
= x y x y = e
p r e m u l t i p l i c a n d o d i c h a i g u a l d a d p o r x y p o s t m u l t i p l i c a n d o p o r y t e n e m o s
x y x y = e
x x y x y y = x e y
e y x e = x y
y x = x y
l u e g o e l g r u p o e s c o n m u t a t i v o .
3 E n c o n t r a r t o d o s l o s s u b g r u p o s n o r m a l e s d e S
3
S o l u c i o n :
S
3
= f i g
1
g
2
s
1
s
2
s
3
g c o n
i =
1 2 3
1 2 3
g
1
=
1 2 3
3 1 2
g
2
=
1 2 3
2 3 1
s
1
=
1 2 3
1 3 2
s
2
=
1 2 3
3 2 1
s
3
=
1 2 3
2 1 3
C o m p o n i e n d o d e t o d a s l a s f o r m a s p o s i b l e s e s t o s e l e m e n t o s , d e d o s e n d o s , o b t e n e m o s
l a s i g u i e n t e t a b l a
i g
1
g
2
s
1
s
2
s
3
i i g
1
g
2
s
1
s
2
s
3
g
1
g
1
g
2
i s
3
s
1
s
2
g
2
g
2
i g
1
s
2
s
3
s
1
s
1
s
1
s
2
s
3
i g
1
g
2
s
2
s
2
s
3
s
1
g
2
i g
1
s
3
s
3
s
1
s
2
g
1
g
2
i
( N o t a : e n l a t a b l a x y e s x c o l u m n a , y l a )
T e n i e n d o e n c u e n t a q u e
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G r u p o s 1 3 5
a ) e l g r u p o e s d e o r d e n s e i s ,
b ) e l o r d e n d e s u s s u b g r u p o s h a d e s e r d i v i s o r d e s e i s ,
c ) u n s u b c o n j u n t o d e u n g r u p o n i t o e s s u b g r u p o s i e s t e e s c e r r a d o c o n r e s p e c t o l a
o p e r a c i o n ,
s i m p l e m e n t e o b s e r v a n d o l a t a b l a a n t e r i o r , v e m o s c u a l e s s o n l o s s u b g r u p o s d e S
3
,
S u b g r u p o s d e o r d e n 1 : f i g
S u b g r u p o s d e o r d e n 2 : f i s
1
g f i s
2
g f i s
3
g
S u b g r u p o s d e o r d e n 3 : f i g
1
g
2
g
S u b g r u p o s d e o r d e n 6 : S
3
S o n s u b g r u p o s n o r m a l e s l o s s u b g r u p o s f i g S
3
( l o s i m p r o p i o s ) , a s c o m o e l d e n d i c e
d o s , q u e p u e s t o q u e e l o r d e n d e S
3
e s s e i s , e s t e e s f i g
1
g
2
g . A n a l i c e m o s s i a l g u n
s u b g r u p o d e o r d e n d o s e s n o r m a l , e s t u d i e m o s p o r e j e m p l o f i s
1
g ( l o s o t r o s d o s s e
e s t u d i a n d e l a m i s m a f o r m a ) .
S e t r a t a d e c o m p a r a r a f i s
1
g , c o n f i s
1
g a c o n a 2 S
3
u n e l e m e n t o c u a l q u i e r a :
s e a a = s
2
s
2
f i s
1
g = f s
2
g
1
g
f i s
1
g s
2
= f s
2
g
2
g
c o n j u n t o s d i s t i n t o s p o r l o q u e e l s u b g r u p o n o e s n o r m a l , ( n i n g u n o d e l o s t r e s s u b g r u p o s
d e o r d e n d o s e s n o r m a l ) .
4 S e a S u n s u b g r u p o d e u n g r u p o G y s e a x 2 G . P r o b a r q u e
x
; 1
S x = f x
; 1
y x 8 y 2 S g
e s u n s u b g r u p o d e G
S o l u c i o n :
S e a n y
1
y
2
2 S e n t o n c e s x
; 1
y
1
x y x
; 1
y
2
x s o n d o s e l e m e n t o s d e x
; 1
S x , v e a m o s s i
s e v e r i c a l a c o n d i c i o n d e s u b g r u p o :
( x
; 1
y
1
x ) ( x
; 1
y
2
x ) = ( x
; 1
y
1
x ) ( x
; 1
y
; 1
2
x ) = x
; 1
y
1
( x x
; 1
) y
; 1
2
x = x
; 1
y
1
y
; 1
2
x
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1 3 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
l a s i g u a l d a d e s a n t e r i o r e s s o n t o d a s e l l a s c i e r t a s p u e s t o q u e x y
1
y
2
s o n e l e m e n t o s d e
G q u e t i e n e e s t r u c t u r a d e g r u p o .
A h o r a b i e n , p o r s e r S s u b g r u p o y
1
y
; 1
2
= y
3
2 S , l u e g o
( x
; 1
y
1
x ) ( x
; 1
y
2
x )
; 1
= x
; 1
y
3
x 2 x
; 1
S x
y p o r l o t a n t o x
; 1
S x e s u n s u b g r u p o d e G
5 S e a A 2 M
2
( C ) y s e a S = f X 2 G L
2
( C ) X A = A X g
a ) > E s S u n s u b g r u p o d e G L
2
( C ) ?
b ) D e t e r m i n a r S p a r a e l c a s o e n q u e A =
0 1
0 0
S o l u c i o n :
a ) S e a n X
1
X
2
2 S l u e g o v e r i c a n X
1
A = A X
1
y X
2
A = A X
2
P a r a v e r s i s e v e r i c a X
1
X
; 1
2
A = A X
1
X
; 1
2
( c o n d i c i o n d e s u b g r u p o ) , v e a m o s p r i m e r o
q u e , s i X
2
2 S e n t o n c e s X
; 1
2
2 S
E n e f e c t o : p r e m u l t i p l i c a n d o y p o s t m u l t i p l i c a n d o l a i g u a l d a d X
2
A = A X
2
p o r X
; 1
2
t e n e m o s
X
; 1
2
X
2
A X
; 1
2
= X
; 1
2
A X
2
X
; 1
2
A X
; 1
2
= X
; 1
2
A
X
; 1
2
A = A X
; 1
2
y n a l m e n t e
X
1
X
; 1
2
A =
( a )
X
1
A X
; 1
2
=
( b )
X
1
A X
; 1
2
( a ) X
; 1
2
2 S
( b ) X
1
2 S
L u e g o e n e f e c t o S e s s u b g r u p o .
b ) S e a X =
x
1
x
2
x
3
x
4
2 S e n t o n c e s
x
1
x
2
x
3
x
4
0 1
0 0
=
0 1
0 0
x
1
x
2
x
3
x
4
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G r u p o s 1 3 7
)
0 x
1
0 x
3
=
x
3
x
4
0 0
)
; x
3
x
1
; x
4
0 x
3
=
0 0
0 0
) x
1
= x
4
x
3
= 0
) X =
x
1
x
2
0 x
1
a h o r a b i e n X 2 G L
2
( C ) l u e g o x
1
6= 0
S =
x
1
x
2
0 x
1
x
1
6= 0
6 P r o b a r q u e ( R ) c o n a b =
3
p
a
3
+ b
3
e s u n g r u p o i s o m o r f o a ( R + )
S o l u c i o n :
V e a m o s q u e ( R ) e s u n g r u p o a b e l i a n o .
1 ) L a o p e r a c i o n e s t a b i e n d e n i d a ( a b e x i s t e p a r a t o d o a b 2 R y e s u n i c o )
A s o c i a t i v i d a d
( a b ) c = (
3
p
a
3
+ b
3
) c =
3
q
(
3
p
a
3
+ b
3
)
3
+ c
3
=
=
3
p
( a
3
+ b
3
) + c
3
=
3
p
a
3
+ ( b
3
+ c
3
) =
3
q
a
3
+ (
3
p
b
3
+ c
3
)
3
=
= a (
3
p
b
3
+ c
3
) = a ( b c )
E x i s t e n c i a d e e l e m e n t o n e u t r o
S i e 2 R e s t a l q u e 8 a 2 R
a e = e a = a
e n t o n c e s
3
p
a
3
+ e
3
= a ) a
3
+ e
3
= a
3
) e
3
= 0
p o r l o t a n t o e = 0 y e v i d e n t e m e n t e e s u n i c o
E x i s t e n c i a d e e l e m e n t o s i m e t r i c o
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1 3 8
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S i p a r a c a d a a 2 R e x i s t e a
1
2 R t a l q u e
a a
; 1
= a
1
a = 0
e n t o n c e s
0 =
3
q
a
3
+ a
3
1
) a
3
= ; a
3
1
) a = a
1
l u e g o e l e l e m e n t o s i m e t r i c o e x i s t e y e s u n i c o
C o n m u t a t i v i d a d
a b =
3
p
a
3
+ b
3
=
3
p
b
3
+ a
3
= b a
L u e g o e n e f e c t o e s g r u p o a b e l i a n o , e s t a b l e z c a m o s a h o r a e l i s o m o r s m o c o n ( R + )
' : ( R ) ; ! ( R + )
a ; ! ' ( a ) = a
3
D i c h a a p l i c a c i o n e s t a b i e n d e n i d a y a q u e ' ( a ) e s u n n u m e r o r e a l u n i c o , p a r a c a d a
a 2 R
E s i n y e c t i v a p u e s ' ( a ) = ' ( b ) ) a
3
= b
3
l o q u e i m p l i c a a = b
E s a d e m a s e x h a u s t i v a p u e s 8 a 2 R e x i s t e
3
p
a t a l q u e ' (
3
p
a ) = a
E s t a a p l i c a c i o n e s m o r s m o d e g r u p o s , y a q u e
' ( a b ) = ' (
3
p
a
3
+ b
3
) = (
3
p
a
3
+ b
3
)
3
= a
3
+ b
3
= ' ( a ) + ' ( b )
p o r l o q u e ' e s u n i s o m o r s m o .
7 S e a G u n g r u p o . P r o b a r q u e s i e x i s t e u n n u m e r o e n t e r o n t a l q u e ( a b )
n
= a
n
b
n
p a r a t o d o a b 2 G e n t o n c e s
G
n
= f x
n
x 2 G g y G
n
= f x 2 G x
n
= e g
s o n s u b g r u p o s n o r m a l e s d e G , y s i G e s u n g r u p o n i t o e n t o n c e s e l o r d e n d e G
n
c o i n c i d e c o n e l n d i c e d e G
n
S o l u c i o n :
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G r u p o s 1 3 9
C o n s i d e r e m o s l a a p l i c a c i o n
' : G ; ! G
x ; ! x
n
y c o m p r o b e m o s q u e e s m o r s m o d e g r u p o s
' ( a b ) = ( a b )
n
=
( a )
a
n
b
n
= ' ( a ) ' ( b )
( a ) p o r h i p o t e s i s
K e r ' = f x 2 G ' ( x ) = e g = G
n
l u e g o G
n
e s s u b g r u p o n o r m a l d e G
I m ' = f y 2 G 9 x 2 G t a l q u e ' ( x ) = y g = f x
n
x 2 G g = G
n
l u e g o G
n
e s u n
s u b g r u p o d e G , v e a m o s q u e t a m b i e n e s n o r m a l
8 y 2 G y x
n
y
; 1
= ( y x y
; 1
)
n
2 G
n
Y p o r u l t i m o ' ( G ) = G
n
' G = G
n
, p o r l o q u e
o r d G
n
= i n d G
n
8 P r o b a r q u e u n g r u p o ( G ) e s a b e l i a n o s i y s o l o s i l a a p l i c a c i o n ' : G ; ! G
d e n i d a p o r ' ( x ) = x
; 1
e s u n a u t o m o r s m o d e G
S o l u c i o n :
L a a p l i c a c i o n ' e s t a b i e n d e n i d a p u e s t o q u e c a d a e l e m e n t o d e G a d m i t e u n i n v e r s o
y e s t e e s u n i c o .
S u p o n g a m o s a h o r a q u e ' e s u n a u t o m o r s m o
' ( a b ) = ' ( a ) ' ( b ) 8 a b 2 G
P o r d e n i c i o n d e ' t e n e m o s
( a b )
; 1
= a
; 1
b
; 1
( 1 )
P o r d e n i c i o n d e e l e m e n t o s i m e t r i c o t e n e m o s
( a b )
; 1
= b
; 1
a
; 1
( 2 )
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1 4 0
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
q u e p o r ( 1 ) y ( 2 )
' ( a b ) = b
; 1
a
; 1
= ' ( b ) ' ( a ) = ' ( b a )
y p o r s e r ' a u t o m o r s m o e s
a b = b a
L u e g o G e s c o n m u t a t i v o .
R e c p r o c a m e n t e
L a a p l i c a c i o n ' e s b i y e c t i v a p o r s e r G g r u p o ( p a r a c a d a e l e m e n t o a 2 G e x i s t e
s i m e t r i c o y e s u n i c o ) , v e a m o s q u e e l h e c h o d e s e r e l g r u p o a b e l i a n o n o s a s e g u r a q u e '
e s m o r s m o
' ( a b ) = ( a b )
; 1
= b
; 1
a
; 1
=
( a )
a
; 1
b
; 1
= ' ( a ) ' ( b )
( a ) h i p o t e s i s d e c o n m u t a t i v i d a d
9 S e a G u n g r u p o . P r o b a r q u e e l o r d e n d e u n e l e m e n t o a 2 G e s e l m i s m o q u e e l
o r d e n d e s u i n v e r s o a
; 1
S o l u c i o n :
S e a n = o r d a e l o r d e n d e a 2 G e s d e c i r a
n
= e . P u e s t o q u e t o d o e l e m e n t o a 2 G
c o n m u t a c o n s u i n v e r s o a
; 1
y e s t e e s t a l q u e a a
; 1
= e , s e t i e n e
( a a
; 1
)
n
= e
n
= e
( a a
; 1
)
n
= a a
; 1
: : : a a
; 1
= a
n
( a
; 1
)
n
y p o r l o t a n t o
a
n
( a
; 1
)
n
= e
A h o r a b i e n a
n
= e l u e g o ( a
; 1
)
n
= e ( a
; 1
)
n
= e . P o r l o t a n t o s i m e s e l o r d e n d e a
; 1
s e t i e n e q u e m e s d i v i s o r d e n
A n a l o g a m e n t e t e n e m o s ( a
; 1
a )
m
= e
m
= e d e d o n d e
a
m
= e a
m
= ( a
; 1
)
m
a
m
= e
p o r l o q u e n e s u n d i v i s o r d e n
F i n a l m e n t e s i n e s d i v i s o r d e m y m e s d i v i s o r d e n e s q u e n = m
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1 4 2
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
a ) ( a ) \ ( b ) = I
S e a n x y 2 I v e a m o s s i x ; y 2 I D e x y 2 I s e t i e n e
x y 2 ( a ) d e d o n d e x ; y 2 ( a )
x y 2 ( b ) d e d o n d e x ; y 2 ( b )
D e x ; y 2 ( a ) , y x ; y 2 ( b ) s e t i e n e x ; y 2 ( a ) \ ( b ) = I
S e a n x 2 I , m 2 Z v e a m o s s i m x 2 I
D e x 2 I s e t i e n e
x 2 ( a ) l u e g o m x 2 ( a )
x 2 ( b ) l u e g o m x 2 ( b )
D e m x 2 ( a ) , y m x 2 ( b ) s e t i e n e m x 2 I
( b ) C o n s i d e r e m o s l o s i d e a l e s I
1
= ( 3 ) , I
2
= ( 2 ) y s e a ( 3 ) ( 2 ) .
T e n e m o s q u e 9 2 ( 3 ) , 4 2 ( 2 ) y 9 ; 4 = 5 = 2 ( 3 ) ( 2 ) p u e s t o q u e 5 = 2 ( 3 ) y 5 = 2 ( 2 ) ,
l u e g o ( 3 ) ( 2 ) n o e s i d e a l .
3 P r o b a r q u e m c d ( a b ) = d , s i e n d o d e l g e n e r a d o r d e l i d e a l s u m a d e l o s i d e a l e s d e
Z g e n e r a d o s p o r a b r e s p e c t i v a m e n t e .
S o l u c i o n :
R e c o r d e m o s q u e
I + J = f a + b a 2 I b 2 J g
e s s i e m p r e u n i d e a l .
E n Z s a b e m o s q u e l o s i d e a l e s s o n p r i n c i p a l e s , l u e g o
( a ) + ( b ) = ( d )
V e a m o s q u e d e s e n e f e c t o m c d ( a b )
( a ) ( d ) p u e s 8 m 2 ( a ) m + 0 = m 2 ( a ) + ( b ) = ( d )
P o r e l m i s m o r a z o n a m i e n t o ( b ) ( d )
D e ( a ) ( d ) t e n e m o s q u e a 2 ( d ) , l u e g o a = d k
1
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A n i l l o s d e c l a s e s d e r e s t o s 1 4 3
D e ( b ) ( d ) t e n e m o s q u e b 2 ( d ) , l u e g o b = d k
2
d e d o n d e d e s d i v i s o r c o m u n d e a , y b f a l t a v e r q u e e s e l m a x i m o .
D e ( d ) = ( a ) + ( b ) t e n e m o s q u e d 2 ( a ) + ( b ) l u e g o e x i s t e n m n 2 Z t a l e s q u e
a m + b n = d
p o r l o q u e s i d
1
e s d i v i s o r d e a y b l o e s d e a m + b n , e s d e c i r , l o e s d e d , p o r l o
q u e m c d ( a b ) = d
4 P r o b a r q u e m c m ( a b ) = c , s i e n d o c 2 Z e l g e n e r a d o r d e l i d e a l i n t e r s e c c i o n d e l o s
i d e a l e s g e n e r a d o s p o r a b 2 Z
S o l u c i o n :
T e n e m o s , p o r h i p o t e s i s , q u e ( a ) \ ( b ) = ( c ) v e a m o s q u e c = m c m ( a , b ) .
D e ( a ) \ ( b ) = ( c ) t e n e m o s
( c ) ( a ) l u e g o c 2 ( a )
( c ) ( b ) l u e g o c 2 ( b )
d e d o n d e
c = a k
1
c o n k
1
2 Z
c = b k
2
c o n k
2
2 Z
l u e g o c e s m u l t i p l o d e a y b . V e a m o s q u e e s e l m n i m o . S e a h u n m u l t i p l o d e a y b
c u a l q u i e r a
h = a h
1
d e d o n d e h 2 ( a )
h = b h
2
d e d o n d e h 2 ( b )
y p o r t a n t o , h 2 ( a ) \ ( b ) , e s d e c i r h = c h
3
, e s t a m b i e n m u l t i p l o d e c
5 P r o b a r q u e p a r a q u e Z = ( n ) s e a c u e r p o , e s c o n d i c i o n n e c e s a r i a y s u c i e n t e q u e n
s e a p r i m o .
S o l u c i o n :
S u p o n g a m o s q u e Z = ( n ) e s c u e r p o , e s d e c i r 8 a 2 Z = ( n ) , a 6= 0 , e x i s t e b 2 Z = ( n ) t a l
q u e a b = 1
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1 4 4
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
S i n n o f u e r a p r i m o , e x i s t i r a n n
1
n
2
2 N a m b o s d i s t i n t o s d e n , t a l e s q u e n
1
n
2
= n
p o r l o t a n t o n
1
n
2
= n = 0
P u e s t o q u e n
1
6= 0 y Z = ( n ) p o r h i p o t e s i s e s c u e r p o , e x i s t e m t a l q u e n
1
m = 1 , l u e g o
0 = 0 m = n
1
n
2
m = n
1
m n
2
= 1 n
2
= n
2
, l u e g o n
2
= n y p u e s t o q u e n
2
n ,
s e t i e n e n
2
= n c o n t r a d i c c i o n l u e g o n h a d e s e r p r i m o , y l a c o n d i c i o n e s n e c e s a r i a .
V e a m o s q u e e s t a m b i e n s u c i e n t e :
S e a 0 6= a = f m n + a 8 m 2 Z c o n 0 < a < n g . P u e s t o q u e n e s p r i m o , m c d ( a n ) = 1
p o r l o q u e e x i s t e n r s 2 Z , t a l e s q u e a r + n s = 1 ( r e c o r d a r q u e ( a ) + ( n ) = ( 1 ) )
l u e g o a r + n s = 1 , o s e a , a r + n s = 1 P e r o n = 0 , p o r l o t a n t o a r = 1 y
Z = ( n ) e s c u e r p o .
6 D e t e r m i n a r t o d o s l o s d i v i s o r e s d e c e r o d e
a ) Z = ( 1 2 ) b ) Z = ( 1 8 ) c ) Z = ( 2 4 ) .
S o l u c i o n :
U n e l e m e n t o a 2 Z = ( n ) c o n a 6= 0 e s u n d i v i s o r d e c e r o s i y s o l a m e n t e s i e x i s t e
b 2 Z = ( n ) , b 6= 0 t a l q u e
a b = 0
O b s e r v a m o s q u e s i a e s d i v i s o r d e c e r o , t a m b i e n l o e s b , y a b = n
a ) 1 2 = 2
2
3 , l u e g o l o s d i v i s o r e s d e c e r o s o n 2 , 3 , 4 , 6 , 8 , 9 , 1 0 e s d e c i r , l a s
c l a s e s d e r e s t o d e l o s d i v i s o r e s p r o p i o s d e 1 2 y d e l o s e l e m e n t o s q u e t i e n e n u n f a c t o r
q u e l o e s d e 1 2 .
O b s e r v a m o s q u e 2 6 = 0 , 3 4 = 0 , 3 8 = 0 , e t c .
b ) 1 8 = 2 3
2
, l u e g o l o s d i v i s o r e s d e c e r o s o n 2 , 3 , 4 , 6 , 8 , 9 , 1 0 , 1 2 , 1 4 ,
1 5 , 1 6 , e s d e c i r , l a s c l a s e s d e r e s t o d e l o s d i v i s o r e s p r o p i o s d e 1 8 y d e l o s e l e m e n t o s
q u e t i e n e n u n f a c t o r q u e l o e s d e 1 8 .
O b s e r v a m o s q u e 2 9 = 0 , 4 6 = 0 , 8 3 = 0 , 4 9 = 0 , e t c .
c ) 2 4 = 2
3
3 , l u e g o l o s d i v i s o r e s d e c e r o s o n 2 , 4 , 8 , 3 , 6 , 1 0 , 1 2 , 1 4 , 1 6 ,
1 8 , 2 0 , 2 2 , 9 , 1 5 , 2 1 , e s d e c i r , l a s c l a s e s d e r e s t o d e l o s d i v i s o r e s p r o p i o s d e 2 4 y
d e l o s e l e m e n t o s q u e t i e n e n u n f a c t o r q u e l o e s d e 2 4 .
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A n i l l o s d e c l a s e s d e r e s t o s 1 4 5
O b s e r v a m o s q u e 2 1 2 = 0 , 4 6 = 0 , 8 3 = 0 , 2 1 8 = 0 , e t c .
7 E s c r i b i r l a s t a b l a s d e s u m a r y m u l t i p l i c a r d e Z = ( 4 ) y r e s o l v e r e l s i s t e m a d e e c u a -
c i o n e s
2 x + 3 y = 1
2 x + 2 y = 1
S o l u c i o n :
+ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1
2 x + 3 y = 1
2 x + 2 y = 1
( a )
) 0 x + y = 2 ) y = 2
( b )
) 2 x + 3 2 = 1 ) 2 x + 2 = 1 ) 2 x = 3
n o t i e n e s o l u c i o n p u e s n o e x i s t e n i n g u n e l e m e n t o x e n Z = ( 4 ) t a l q u e 2 x = 3 . O b s e r v e s e
q u e 2 n o e s i n v e r s i b l e e n Z = ( 4 ) ( e s u n d i v i s o r d e c e r o )
( a ) s u m a n d o a m b a s e c u a c i o n e s .
( b ) s u s t i t u y e n d o e l v a l o r d e x e n l a p r i m e r a e c u a c i o n
8 E s c r i b i r l a t a b l a d e s u m a r y m u l t i p l i c a r d e l c u e r p o Z / ( 5 ) y r e s o l v e r e l s i s t e m a
x + 2 y = 1
2 x + y = 0
S o l u c i o n :
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1 4 6
A l g e b r a L i n e a l . P r o b l e m a s r e s u e l t o s
+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3
0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1
x + 2 y = 1
2 x + y = 0
( a )
)
2 x + 4 y = 2
2 x + y = 0
( b )
) 4 x + 0 y = 2
) 4 x = 2 ) x = 3
( c )
) 2 3 + y = 0 ) 1 + y = 0 ) y = 4
( a ) m u l t i p l i c a n d o l a p r i m e r a e c u a c i o n p o r 2 .
( b ) s u m a n d o a m b a s e c u a c i o n e s .
( c ) s u s t i t u y e n d o e l v a l o r d e x e n l a s e g u n d a e c u a c i o n
9 D e s c o m p o n e r e n f r a c c i o n e s s i m p l e s , s o b r e Z / ( 5 ) l a f r a c c i o n r a c i o n a l s i g u i e n t e
4
x
2
+ 4 x + 3