Apuntes de Algebra 2 ma - 290 corregido -2- -1 PDF

Title Apuntes de Algebra 2 ma - 290 corregido -2- -1
Course Álgebra II
Institution Universidad Católica del Norte
Pages 107
File Size 2.2 MB
File Type PDF
Total Downloads 670
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Summary

DEPARTAMENTO DE MATEMÁTICAS ALGEBRA II MA-290 Tomás Ávila y Mario Salas Antofagasta, Agosto 2012. 1 1 Espacios vectoriales 1 Espacio vectorial Un espacio vectorial sobre un cuerpo K (en general R o C) es un conjunto V = ∅ sobre el que hay definidas dos operaciones : 1. Suma : + : V × V −→ (u, v) −→...


Description

  

  

    

  



     

K

! "#$ "$ %" & $&"# V

 ∅ 

R

'" ""

C



( " & $)&

%" " *&" + ,"-, , #"$" .

 



.

×

V

−→ −→

V

, 

/



"-$,  &"" ##",," .

'( & . '%( $ .

0"

∈V

V

 ,

 −  −    





∀  ∈ ,

"0"

V

,

V

        

 *&"

V

',( "" #&" .



,

           ,

'$( "" "& .

∀ ∈

∀ ∈

      ,

− ∈

V

 *&"

   !  



.



−→ −→

V

λ, 

V λ



"-$,  &"" ##",," .

'(

    ,

'%( λ '$(

V

         ∀ ∈K ∀∈      ∀ ∈K ∈           ∀ ∈ K ∀   ∈

λ

',( λ

∀∈





λ λ

λ

,

u

λ, 



λ

u,

,

V

λ, 

,

λ

, u

,

,

V V

    " & "#$ "$ %" " $&"# R ,"

1 """ ," & "#$ "$ "  !

 2" ""



" 

  &$" "#" *&"  + $&3 "  " #&

"  #"$3 #,&$ # "$



T. Ávila y M. Salas

#   "#$ "$" "" $  #"$" *&" " ,$  &"" .



 $)& ,"

R

n





x

n4&# ," 2" "" .

 x , x , ..., xn   x i ≤i≤n  xi

∈R

≤i≤n

,



$  #"$" .

    x  y , x  y , ..., xn  yn    λx , λx , ..., λx n 

λ



 $)& ," $" ," ,"3

M

n×m

R  {A  a ij  a ij

$  #"$" . 5

∈R

n

×

,

 ≤ i ≤ n,  ≤ j

m .

≤ } m

& ," $"  #,&$ # 2" ""

 $)& ," ,  # $ $"-$"" "" "  %" x .

P R 

 p

x 

n 

ak x

k

 n ∈ N,

ak

∈R



k 

$  $$ #"$" ," &  #,&$ # 2" "" 6

 $)& ," ,  # $ $"-$"" "" "  4 %"

x ," , "  & *&"

P R  n

  

n .

n 

p x

ak x

k

 ak ∈ R



k 

$   #"$" "" 7

 $)& ," ,  &$" "" .

F R  {  R −→ R} f

$  #"$" .

& ," &$"  #,&$ # 2" ""

5

T. Ávila y M. Salas

8

 $)& ," ,  &$"" ," 2" "" .

S  {xn ∞ n  xn ∈ R, $  #"$" .

n

≥ }

& ," &$""  #,&$ # 2"

"" 9

Z  { }



Z

,

, "$"

Z n

" & "#$ "$ %" " $&"#

 $  #"$" .

        ,     

                 



     V " & "#$ "$ "$"



  



−    −

# ,

∈V

$ %  " 

%  ," & "#$ "$

$)&  $: S

⊂V

,"-, %"

" ," # .

V

V

 $&*&" &%4

*&" " "#$ "$ $   #"$" S



V

& '()!  %   V " & "#$ "$  S

S

" &%"#$ "$ ,"

V

⊂ V  S   "$" ⇐⇒



'( '(

   ∈ S, ∀ ,  ∈ S  ∈ S, ∀ λ ∈ K  ∀  ∈ S

λ

6

T. Ávila y M. Salas

*)! 

⇒ ,"" #&" S " "#$ "$ ⇐    ; *&"  #"$" " %" ,"-, %" S   " , ∀ f x| ≤ ,

∈ R} ∀ ∈ R} x

 &%"#$ "$" ," "#$ "$ # " S

x

x

P R

 ," , 

$ $"-$"" ""  &"" .

 {p ∈ P R  p ′   }

S

 {p ∈ P R  a  a  }

7

T. Ávila y M. Salas

,,"



a

a

  $"-$"" ," ,

  &%"#$ "$" .

 

S S

6

{ ∈ P R  { ∈ P R 





 "#"$""

}

}

p x  

p

gr

p

" , ," p x

 

" #

 " "#$ "$ ," ,  $" $&,, ," ," n " &%$)& ,"  $" 

"

T



T

− S  ,," S

R  −→ R 

$3$

 { , ,  }  %"  K erT   ,   −  ∈  T , T   T

T

"#"$ ,"

B

 { , ,   } ",      − ,    −     "#"$ ,"

B



'( B  ; ,"

T



 Gen {, , − , −, , }

& #$$3 " 

%", *&"

 *&"  ; ,"

T

"#"$ ,"

B

$$," $  ; ," B



B

6>

T. Ávila y M. Salas

 %" ,"

R

T

 *&"

'%( B  "$&$" #:$ ,"

5

"

T



M

R

× 

−→ P R 



 $3 " ,"-, #



a

b

c

d

'( B  ; ,"

T

'%( B & %" ,"

 T 

T

'$( B & %" ,"

M

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{

,"

S

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M

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−→− −

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# T

U

T



  , M     ," M  

M , M

}

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"

$ & ," & ," S

 +  $,",

S

"#"$ ," ,$+ %"

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K erT  & &#""



W

 T

 & &#""

  "$%"  ;

  ,"

P R

"#"$ ,"  %" &&"

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 T

 ,"

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T

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P

   

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'%( '%4(

'%4(

 $&*&"

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=



D



  

      

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 

a

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D

 

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      

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 ,;%"  "

>

∈R

a

B

 {  x , x  x ,  − x , x − x } 

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       −     −

  

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

$ m.a.

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  Gen {, , , }

a, b, c, d  −c, a  b − d, −c, c

'%( B

KerT 



−   

 ,

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',( B





−  − 

 ,

−   

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b−d  c  

T   {−, , , , }



σ

,

   



$& " ,



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T. Ávila y M. Salas

,

   





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   B   

−  − 

 

 −  5

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 n 

'%( A

 n   

'( T

x, y, z   y  z , x  y  z, x  y 

'%( B



 n −     n   −  − n



 − − −  − − − 

 

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a − b

  n   

      

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a

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6

7

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λ   λ  λ

S

Gen

{, ,  , , , }



S

λ   

Gen

{, , }

$





8

' %(

9

σ

S

T  

 Gen {, , , − , , , , }

  





,

 

T B  

   

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a a

 d  ,  



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D





P

a



I

d   

 

P



 −

− a −

d

 d  , M n  I    a    d    "$"  n  a  d −  a −   d −  − a  d −  n d −   d −  n M  n − a  d −  a −   a −  a  d −  d −   a −  a  d−

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a

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T. Ávila y M. Salas



&  .  &     .  " 

   %" & "#$ "$ "

#$$3 <

 ,

>

× −→ R   −→  

V

V

,



 $&*&"

V

<

,

>

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< ,  >< ,  >, ∀ ,  ∈ V , λ   > λ < ,  >  < ,  > < λ  ,  > λ < ,  >  < ,  >

4 & . 4 E" .

∀  ∈ ,

,

<

V,



∈ R

λ, 

<

4 "-, # .

<

,

,  > ≥ , ∀  ∈ V ,  >  ⇐⇒   

! "#$ "$ $ & #,&$ "$ ,"-, %"  

T

   

 *&" " .

4 & .

<

T

,  >  

  x

y



      

x y



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T. Ávila y M. Salas

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T



<

,  > T < ,  >

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 

λ   

λ"

, # 

 #  ##",, $& " "-$ %

T. Ávila y M. Salas

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!;)! 0!

 & ",-

V, < ,  >

%" & "#$ "$ "&$:,"

T



−→

V

V

*&" $" " #,&$

"$ " ,"$ *&"

"T

 , T #  ", # ,

+ ( 0! ! ; $&,,

t A A  I

A

+$ # 

1 ;

 A  √ 

 T A A  √



1 ;

 



 A  √ 

 T A A  √



R

∈ M n×n 





 

−

 

−

( 0! 

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 

−





  



   

 I

  "  #&"



  − 

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,  ∈ V





  − 





 √ 



 

−





  



   

   I

+& 2 "

A

 T  B

 ; ,"  $3 

"#"$ ," & %" 

1 $3

T

B

,"

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V, <

⇐⇒

, 

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T. Ávila y M. Salas

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V

−→

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" 

V

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 , T #  T  ", #   

"T

T

T

  A

T

T



A

A

T



A

T

," ,," " ,",&$" *&" T

A

A

 I

" ,"$   3 

++ 4%)!  T

A

 a ≤ A  c  ≤ A

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T

ij

i,j

ij

i,j

≤n

≤n

A

R

∈ Mn×n 



" & ; 

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