apostila_Álgebra_Linear_2006

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Sumário
1 Espaços Vetoriais 7
 ! "#$%&'()*+,& - ./-012&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4
 !5 6%&1%7-','-3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8
 !8 ./-%*9:*7&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ;
2 Subespaços Vetoriais 17
5! "#$%&'()*+,& - ./-012&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4
5!5 "#$-%3-)*+,& - <&0, '- <(=-31,)*&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 5>
5!8 ./-%*9:*7&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 5;
3 Combinações Lineares 29
8! "#$%&'()*+,& - ./-012&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 5?
8!5 @-%,'&%-3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8>
8!8 ./-%*9:*7&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8;
4 Dependência Linear 37
A! "#$%&'()*+,& - ./-012&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 84
A!5 6%&1%7-','-3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A 
A!8 ./-%*9:*7&3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A8
5 Base, Dimensão e Coordenadas 45
;! B,3- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A;
;!5 C70-#3+,& ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A4
8
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#
$%&'
!"# $%&'()*+, -' .,&+ -' ./0')1+23,) 4'5,6%+%) " " " " " " " " " !7
!" 8,,6-'(+-+) " " " " " " " " " " " " " " " " " " " " " " " " " " !9
!"! :;'63<=3%,) " " " " " " " " " " " " " " " " " " " " " " " " " " " " !>
6 Mudança de Base 61
9"7 ?(56,-/23*+,@ :;'&1A,) ' B6,16%'-+-') " " " " " " " " " " " " " 97
9"C :;'63<=3%,) " " " " " " " " " " " " " " " " " " " " " " " " " " " " 9D
7 Exerćıcios Resolvidos – Uma Revisão 71
8 Transformações Lineares 85
>"7 ?(56,-/23*+, ' :;'&1A,) " " " " " " " " " " " " " " " " " " " " " >!
>"C E :)1+23, 4'5,6%+A L (U,V) " " " " " " " " " " " " " " " " " " >>
>"# ?&+F'& ' G</3A', " " " " " " " " " " " " " " " " " " " " " " " " H!
>" ?),&,6I)&, ' J/5,&,6I)&, " " " " " " " " " " " " " " " " " 7K#
>"! L+56%M -' /&+ N6+()O,6&+23*+, P%('+6 " " " " " " " " " " " " " 7K9
>"!"7 $'I(%23*+, ' :;'&1A,) " " " " " " " " " " " " " " " " " 7K9
>"!"C B6,16%'-+-') " " " " " " " " " " " " " " " " " " " " " " 7K>
>"9 :;'63<=3%,) Q'),AR%-,) " " " " " " " " " " " " " " " " " " " " " " 77#
>"D :;'63<=3%,) " " " " " " " " " " " " " " " " " " " " " " " " " " " " 77H
9 Autovalores e Autovetores 127
H"7 $'I(%23*+,@ :;'&1A,) ' B6,16%'-+-') " " " " " " " " " " " " " 7CD
H"C B,A%(S,&%, 8+6+35'6<=)5%3, " " " " " " " " " " " " " " " " " " " " 7#!
H"# :;'63<=3%,) " " " " " " " " " " " " " " " " " " " " " " " " " " " " 7#H
10 Diagonalização 141
7K"7 $'I(%23*+, ' 8+6+35'6%M+23*+, " " " " " " " " " " " " " " " " " " " 7 7
7K"C :;'63<=3%,) " " " " " " " " " " " " " " " " " " " " " " " " " " " " 7!#
11 Forma Canônica de Jordan 155
77"7 ?(56,-/23*+, ' :;'&1A,) " " " " " " " " " " " " " " " " " " " " " 7!!
 !"
#
$%&' 
!!"# $%&'()*(+,- " " " " " " " " " " " " " " " " " " " " " " " " " " " " !.#
12 Espaços Euclidianos 163
!#"! /',012, 342&'4, " " " " " " " " " " " " " " " " " " " " " " " " !.5
!#"# 6,'78 " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " !.9
!#"5 :+-2;84(+8 " " " " " " " " " " " " " " " " " " " " " " " " " " " " !9<
!#"=
;
>4?1@, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " !9!
!#" A'2,?,48@+080& " " " " " " " " " " " " " " " " " " " " " " " " " !9#
!#". /',(&--, 0& B'87CD(E7+02 " " " " " " " " " " " " " " " " " " !9F
!#"9 G,7H@&7&42, A'2,?,48@ " " " " " " " " " " " " " " " " " " " " !F5
!#"F 3-,7&2'+8 " " " " " " " " " " " " " " " " " " " " " " " " " " " " !F=
!#"I AH&'80,' >12,C80J142, " " " " " " " " " " " " " " " " " " " " !FI
!#"!<$%&'()*(+,- " " " " " " " " " " " " " " " " " " " " " " " " " " " " !I#
 !"
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Caṕıtulo 1
Espaços Vetoriais
1.1 Introdução e Exemplos
 
 !" #$%&'"()* +,"-*.(/+- 0*! * #*,# +"* . !%$1#* 2 "*-+$) 3( ! -&$
(!$.* 0 "*.* * . #*-- - .* #(-!*4
5*-& 06 $," ! . $%- ! ,"$-0*! $ . 7,+1#8$* . !%$1#* 2 "*-+$)6 %$!! 9
0*! $ $,$)+!$- 0 %$-$) )* .*+! *:; "*!< * #*,;(,"* =*-0$.* % )$! =(,1#8* !
f : R → R6 . ,*"$.* %*- F (R; R) * #*,;(,"* .$! 0$"-+/ ! 3($.-$9
.$! . *-. 0 n #*0 #* 7#+ ," ! - $+! 3( . ,*"$- 0*! %*- Mn(R), *(
!+0%) !0 ," 6 %*- Mn.
> !*0$ . .($! =(,1#8* ! f g . F (R; R) & . 7,+.$ #*0* ! ,.* $
=(,1#8$* f + g ∈ F (R; R) .$.$ %*- (f + g)(x) = f(x) + g(x).
?*" "$0:& 0 3( ! λ ∈ R %*. 0*! 0()"+%)+#$- $ =(,1#8$* f % )* !#$)$-
λ, .$ ! @(+," =*-0$ (λf)(x) = λ(f(x)), - !()"$,.* ,(0 ) 0 ,"* . F (R).
A*0 - )$1#8$* $ Mn %*. 0*! !*0$- .($! 0$"-+/ ! 3($.-$.$! . *-. 0
n, A = (aij)n×n B = (bij)n×n, #*)*#$,.* A + B = (aij + bij)n×n, 3( & 
(0 ) 0 ,"* . Mn.
A*0 $ - )$1#8$* B$ 0()"+%)+#$1#8$* . A = (aij)n×n %*- (0 !#$)$- λ ∈ R,
& ,$"(-$) . 7,+-0*! λA = (λaij)n×n, * 3($) "$0:& 0 % -" ,# $ Mn.
C 3( !" ! .*+! #*,;(,"*! $#+0$6 #*0 !"$! !"#$"$#%! . $.+1#8$* . 
D
 !"
#
$%&'( )* +,"! -(, .+%(/$!$,
!"#! "$"%"&'(! " %#$')*$)+,-+.,( /" !"#! "$"%"&'(! *(0 "!+,$,0"!1 '2"% +(3
%#%4 5"6,%(!7
5"0)8+,3!" 9,+)$%"&'" , *,0')0 /,! *0(*0)"/,/"! /(! &:#%"0(! 0",)! ;#"1
+(% 0"$,-+.,( , ;#,)!;#"0 9#&-+.("! f, g " h "% F (R; R) " *,0, '(/( λ, µ ∈ R,
!.,( <:,$)/(! (! !"=#)&'"! 0"!#$',/(!7
>? f + g = g + f;
@? f + (g + h) = (f + g) + h;
A? !" O 0"*0"!"&', ( 9#&-+.,( &#$,1 )!'( :"1 O(x) = 0 *,0, '(/( x ∈ R
"&'.,( O + f = f;
B? , 9#&-+.,( −f /"8&)/, *(0 (−f)(x) = −[f(x)] *,0, '(/( x ∈ R :" ',$
;#" f + (−f) = O ;
C? λ(µf) = (λµ)f;
D? (λ + µ)f = λf + µf;
E? λ(f + g) = λf + λg;
 ? 1f = f.
F=(0,1 +(% 0"$,-+.,( , ;#,)!;#"0 %,'0)G"! A,B " C "% Mn " *,0, '(/(
λ, µ ∈ R, ',%H:"% !.,( <:,$)/(! (! !"=#)&'"! 0"!#$',/(!7
>? A + B = B + A;
@? A + (B + C) = (A + B) + C;
A? !" O 0"*0"!"&', ( 9#&-+.,( &#$,1 )!'( :"1 O = (0)n×n "&'.,( O + A = A;
B? !" A = (ai,j)n×n "&'.,( , %,'0)G −A /"8&)/, *(0 −A = (−ai,j)n×n :"
',$ ;#" A + (−A) = O;
C? λ(µA) = (λµ)A;
 ! ! "#$%&'()*
+
,& - -.-/01&2 
!" (λ + µ)A = λA + µA;
#" λ(A + B) = λA + λB;
$" 1A = A.
%&'()&* +(, -.( /01/& & 2&13.1/&* '0* 4.1526&(* '(718'0* 10 ,(/0 0 +09&:
,(* ,(08* 2&)& & '0* )0/,8;(* -.0',0'0* -.01'& ).18'&* '( *&)0* ( ).9:
/8<98205260& <&, (*2090,(* 0'(-.0'0* 0<,(*(1/0) !" !#$%&%$' &()*$+!#,&'
2&).1*" =0 +(,'0'( ).8/&* &./,&* 2&13.1/&* ).18'&* '( &<(,0526&(* 0<,&:
<,80'0* 0<,(*(1/0) <,&<,8('0'(* *()(9>01/(* ?0* 028)0"
@
A <&, 8**& -.( 0& 81+@(* '( (*/.'0,)&* 20'0 .) *(<0,0'0)(1/( (*/.'0:
,()&* .) 2&13.1/& 0,B8/,@0,8& ( 160& +0;8&C V, *&B,( & -.09 *.<&)&* (*/0,
'(718'0* .)0 &<(,05260& '( 0'85260&C 8*/& @(C <0,0 20'0 u, v ∈ V (D8*/( .)
@.182& (9()(1/& '( V 0**&280'&C 2>0)0'& 0 *&)0 (1/,( u ( v ( '(1&/0'&
<&, u+v, ( .)0 ).9/8<98205260& <&, (*2090,C 8*/& @(C <0,0 20'0 u ∈ V ( λ ∈ R
(D8*/( .) @.182& (9()(1/& '( V 0**&280'&C 2>0)0'& '( !"%-." %$ u $("
$',&(&! λ ( '(1&/0'& <&, λu.
Definição 1.1 /#!$0"' 1-$ -0 ,"23-2." V ,"0" &,#0& 0-2#%" %$
-0& &%#4,5&" $ %$ -0& 0-(.# (#,&4,5&" "! $',&(&! *$ -0 (*<052& +(/&,809
'$ &!& 1-&#'1-$! u, v $ w $0 V $ &!& ."%" λ, µ ∈ R '5&" 6*&(#%&' &'
'$)-#2.$' !" !#$%&%$'7
 !"#$ u + v = v + u &!& ."%" u, v ∈ V ;
 !"%$ u + (v + w) = (u + v) + w &!& ."%" u, v, w ∈ V ;
 !"&$ $8#'.$ -0 $($0$2." 0 ∈ V .&( 1-$ 0 + u = u &!& ."%" u ∈ V ;
 !"'$ &!& ,&%& u ∈ V $8#'.$ v ∈ V .&( 1-$ u + v = 0;
 !"($ λ(µu) = (λµ)u &!& ."%" u ∈ V $ λ, µ ∈ R;
 !")$ (λ + µ)u = λu + µu &!& ."%" u ∈ V, λ, µ ∈ R;
 ! !"
#
$%&'( )* +,"! -(, .+%(/$!$,
 !"#$ λ(u + v) = λu + λv !"! #$%$ u, v ∈ V & λ ∈ R;
 !"%$ 1u = u !"! #$%$ u ∈ V.
Observação 1.2 '( )$*+* ),!*!"*$- $- &.&*&/#$- %& +* &- !0)$ 1&2
#$"3!. %& 1&#$"&-4 3/%& &/%&/#&*&/#& %! /!#+"&5! %$- *&-*$-6 7!*2
8'&* ),!*!*$- %& &-)!.!"&- $- /'+*&"$- "&!3- 9+!/%$ &-#&- %&-&* &2
/,!* $ -&+ ! &. /! !0):!$ %& *+.#3 .3)!" +* 1&#$"6
Observação 1.3 ; &.&*&/#$ 0 /! "$ "3&%!%& !"& '& '+/3)$4 $3- 9+!.2
9+&" $+#"$ 0 ′ ∈ V -!#3-<!5&/%$ ! *&-*! "$ "3&%!%& !"& &/#:!$4 &.!-
 "$ "3&%!%&- !"& & !"' #&"'=!*$- 0 ′ = 0+0 ′ = 0 ′+0 = 0, 3-#$ '&4 0 = 0 ′.
Observação 1.4(* +* &- !0)$ 1&#$"3!.4 &.! "$ "3&%!%& !"(4 !"!
)!%! u ∈ V &>3-#& v ∈ V #!. 9+& u + v = 0. ?! 1&"%!%&4 !"! )!%!
u ∈ V &>3-#& -$*&/#& +* &.&*&/#$ v ∈ V )$* &-#! "$ "3&%!%&6 @&
<!#$4 %!%$ u ∈ V -& v & v ′ &* V -:!$ #!3- 9+& u + v = 0 & u + v ′ = 0
&/#:!$4 )$*83/!/%$ &-#!- &9+!0):$&- )$* !- "$ "3&%!%&- !"'4!") &
!"&4 $8#&*$- v = v + 0 = v + (u + v ′) = (v + u) + v ′ = (u + v) + v ′ =
0 + v ′ = v ′, 3-#$ '& v = v ′. @&/$#!"&*$- v $" −u & u − v $" u + (−v).
Observação 1.5 A- 9+!#"$ "3*&3"!- "$ "3&%!%&- "&<&"&*2-& ! &/!-
B! $ &"!0):!$ %& !%30):!$ & -:!$ )$/,&)3%!-4 "&- &)#31!*&/#&4 $" "$ "3&2
%!%& )$*+#!#31!4 "$ "3&%!%& !--$)3!#313%!%&4 &>3-#C&/)3! %$ &.&*&/#$
/&+#"$ & &>3-#C&/)3! %$ &.&*&/#$ 3/1&"-$6
A 9+3/#! & ! $3#!1! "$ "3&%!%&- -:!$ &>).+-31!- %! *+.#3 .3)!0):!$
 $" &-)!.!" & #!*8'&* $%&* -&" ),!*!%!- %& !--$)3!#313%!%& & &.&2
*&/#$ /&+#"$ %! *+.#3 .3)!0):!$4 "&- &)#31!*&/#&6
A -&>#! & ! -'&#3*! "$ "3&%!%&- "&.!)3$/!* !- %+!- $ &"!0):$&- &
-:!$ !*8!- )$/,&)3%!- $" %3-#"38+#313%!%&6
Observação 1.6 A "3D$"4 ! %&E/30):!$ %& &- !0)$ 1&#$"3!. 9+& %&*$-
!)3*! -& "&<&"& ! &- !0)$- 1&#$"3!3- "#$%& 13-#$ 9+& &-#!*$- &"*3#3/%$
 ! ! "#$%&'()*
+
,& - -.-/01&2 
 !" #$ "$%&'&("$ $")&* &+",&$ ,-!*"(#$ ("&.$/ 0 ,#1%2&# 3" "$+&1%#
4"5#(.&' !"#$%&'" $"(& )&* +&,-. /.-0*.%#&/-& . $.*-,* (. (&1/,2!3." .!,#.
!"# .) (&4,(.) #0(./2!.)5 6.,) $*&!,).#&/-&7 $&(,#") 80& )&9. ).-,)+&,-.)
.) $*"$*,&(.(&) !" . !# & !$ &/80./-" 80& .) $*"$*,&(.(&) !% .
 !& (&4&# 4.%&* $.*. -"(" λ, µ ∈ C. :" &/-./-"7 &#;"*. ,#$"*-./-&7 /3."
0).*&#") " !"/!&,-" (& &)$.2!" 4&-"*,.% !"#$%&'"5
<# "0-*" &'&#$%" (& &)$.2!" 4&-"*,.%7 .%=&# (") (",) .$*&)&/-.(") /"
,/=>!," (" -&'-"7 =& " !"/90/-" (") 4&-"*&) !"#" .$*&)&/-.(") &# ?&"#&-*,.
@/.%=>-,!. #0/,(" (. .(,2!3." & (. #0%-,$%,!.2!3." $"* &)!.%.*5 A&)). +"*#.7
" .(9&-,4" 4&-"*,.% 0-,%,B.(" /. (&1/,2!3." .!,#. (&4& )&* &/-&/(,(" (& 0#.
+"*#. #.,) .#$%.7 )&/(" 0#. *&+&*C&/!,. .") &%&#&/-") (& V ,/(&$&/(&/D
-&#&/-& (& )&*&# "0 /3." 4&-"*&)5
E.%4&B " &'&#$%" #.,) ),#$%&) (& &)$.2!" 4&-"*,.% )&9. " !"/90/-" (")
/=0#&*") *&.,) !"# . .(,2!3." & #0%-,$%,!.2!3." 0)0.,)5 6.,) F&*.%#&/-&7 $.*.
!.(. n ∈ N, $"(&#") -*./)+"*#.* " !"/90/-" (.) nD0$%.) "*(&/.(.) (&
/=0#&*") *&.,)7 R
n, &# 0# &)$.2!" 4&-"*,.% (&1/,/(" . .(,2!3." (& (0.)
nD0$%.) "*(&/.(.)7 x = (x1, . . . , xn) & y = (y1, . . . , yn), .(,!,"/./("D)&
!""*(&/.(. . !""*(&/.(.7 ,)-" =&7
x + y = (x1 + y1, . . . , xn + yn)
& " $*"(0-" (& 0#. nD0$%. x = (x1, . . . , xn) $"* 0# &)!.%.* λ ∈ R $"*
λx = (λx1, . . . , λxn).
=
G 0#. *"-,/. ;&# ),#$%&) 4&*,1!.* 80& (&))& #"(" R
n
=& 0# &)$.2!" 4&D
-"*,.%5 A&,'.#") !"#" &'&*!=>!," &)-. -.*&+.5
H&*,180& -.#;=&# 80& ") )&F0,/-&) &'&#$%") )3." &)$.2!") 4&-"*,.,)5
 5 I&9.# n ∈ N & V = Pn(R) " !"/90/-" +"*#.(" $&%" $"%,/C"#," /0%" &
$"* -"(") ") $"%,/C"#,") (& F*.0 #&/"* "0 ,F0.% . n !"# !"&1!,&/-&)
*&.,)5 A&1/,#") . .(,2!3." & . #0%-,$%,!.2!3." $"* &)!.%.* (. )&F0,/-&
#./&,*.J
 ! !"
#
$%&'( )* +,"! -(, .+%(/$!$,
 "# p(x) = a0 + a1x + · · ·+ anxn # q(x) = b0 + b1x + · · ·+ bnxn
$%&' #(#)#*+'$ ,# Pn(R) #*+%&'
p(x) + q(x) = (a0 + b0) + (a1 + b1)x + · · · + (an + bn)xn.
 "# p(x) = a0 + a1x + · · · + anxn -# .) #(#)#*+' ,# Pn(R) #
λ ∈ R #*+%&'
λp(x) = (λa0) + (λa1)x + · · · + (λan)xn.
!/ "#0&) A ⊂ R # F (A; R) ' 1'*0.*+' ,# +',&$ &$ 2.*31%'#$ f : A → R.
"# f, g ∈ F (A; R) # λ ∈ R ,#4*& f + g : A → R 5'6 (f + g)(x) =
f(x)+g(x) # (λf)(x) = λf(x), x ∈ A. 7*+%&'8 F (A; R) 1') #$+& &,931%&'
# 56',.+' 5'6 #$1&(&6 -# .) #$5&31' :#+'69&(/
;/ < 1'*0.*+' ,&$ 2.*31%'#$ 1'*+-=*.&$ ,#4*9,&$ *.) 9*+#6:&(' I ⊂ R
).*9,' ,&$ '5#6&31%'#$ ,# &,931%&' # ).(+95(91&31%&' .$.&9$ >1')' &?.#(&$
,#4*9,&$ #) F (I; R)). @'+&31%&'A C(I; R).
B/ < 1'*0.*+' ,&$ 2.*31%'#$ 1') ,#69:&,&$ 1'*+-=*.&$ &+-# '6,#) k ∈ N, >k
-# 4C'D ,#4*9,&$ *.) 9*+#6:&(' &E#6+' I ⊂ R ).*9,' ,&$ '5#6&31%'#$ ,#
&,931%&' # ).(+95(91&31%&' .$.&9$ >1')' &?.#(&$ ,#4*9,&$ #) F (I; R)).
@'+&31%&'A Ck(I; R).
F/ < 1'*0.*+' ,&$ 2.*31%'#$ 1') +',&$ &$ ,#69:&,&$ 1'*+-=*.&$ ,#4*9G
,&$ *.) 9*+#6:&(' &E#6+' I ⊂ R ).*9,' ,&$ '5#6&31%'#$ ,# &,931%&' #
).(+95(91&31%&' .$.&9$ >1')' &?.#(&$ ,#4*9,&$ #) F (I; R)). @'+&31%&'A
C∞(I; R).
H/ < 1'*0.*+' ,&$ )&+69I#$ m 5'6 n 1') 1'#419#*+#$ 6#&9$A Mm×n(R)
).*9,' ,# '5#6&31%'#$ &*-&('J&$ K&?.#(&$ ,#4*9,&$ #) Mn(R).
<$ #$5&31'$ :#+'69&9$ &19)& #*:'(:#) '5#6&31%'#$ 1') &$ ?.&9$ :'1L# 0-&
,#:# #$+&6 2&)9(9&69I&,'/ < 56-'C9)' #C#)5(' -# .) 5'.1' )&9$ $'4$+91&,'
 !"! #$%#$&'()('* !
"# $%& #' ()*&+,#+&' & -#+ ,''# .#'*+(+&.#' (' #,*# -+#-+,&"("&'/ 0#.#
1#)2%)*# *#.(+&.#' V = (0,∞), # '&.,3&,4# -#',*,5# "( +&*( +&(6/ 7'*&
1#)2%)*# $%()"# (8+&8("# 9(' #-&+(:1;#&' %'%(,' "& '#.( & .%6*,-6,1(:1;(#
);(# <& %. &'-(:1# 5&*#+,(6= 5,'*# $%& );(# -#''%, ! " #$% # &$'% -(+( (
(",:1;(#/ ># &)*()*#= '& -(+( x, y ∈ V & λ ∈ R, "&?),+.#' ( '#.( &)*+& x
& y -#+ x ⊞ y = xy, @# -+#"%*# %'%(6 &)*+& x & yA & # -+#"%*# "& x -&6#
&'1(6(+ λ 1#.# λ ⊡ x = xλ, &)*;(# V '& *#+)( %. &'-(:1# 5&*#+,(6/ B& C(*#=
5&+,?$%&.#' %.( ( %.( (' #,*# -+#-+,&"("&'D
 / x, y ∈ V *&.#' x ⊞ y = xy = yx = y ⊞ x -(+( $%(,'$%&+ x, y ∈ V ;
E/ x⊞ (y⊞ z) = x⊞ (yz) = x(yz) = (xy)z = (x⊞y)z = (x⊞y)⊞ z -(+(
$%(,'$%&+ x, y, z ∈ V
!/ '& x ∈ V &)*;(#= 1#.# 1 ∈ V, *&.#' 1⊞x = 1x = x; #F'&+5& $%& )&'*&
1('#= <& # &6&.&)*# )&%*+# "( ()*+,-(%= # $%(6 "&)#*(+&.#' -#+ o;
G/ '& x ∈ V, ,'*# <&= x > 0, &)*;(# x−1 ∈ V & x ⊞ x−1 = xx−1 = 1 = o;
H/ λ ⊡ (µ ⊡ x) = λ ⊡ xµ = (xµ)λ = xµλ = xλµ = (λµ) ⊡ x -(+( $%(,'$%&+
x ∈ V & λ, µ ∈ R;
I/ (λ+µ)⊡x = xλ+µ = xλxµ = xλ⊞xµ = (λ⊡x)⊞(µ⊡x) -(+( $%(,'$%&+
x ∈ V & λ, µ ∈ R;
J/ λ ⊡ (x ⊞ y) = λ ⊡ (xy) = (xy)λ = xλyλ = (λ ⊡ x) ⊞ (λ ⊡ y) -(+(
$%(,'$%&+ x, y ∈ V & λ ∈ R;
K/ 1 ⊡ x = x1 = x -(+( $%(6$%&+ x ∈ V.
1.2 Propriedades
B(' #,*# -+#-+,&"("&' $%& "&?)&. %. &'-(:1# 5&*#+,(6 -#"&.#' 1#)16%,+
5<(+,(' #%*+('/ L,'*(+&.#' (68%.(' "&'*(' -+#-+,&"("&' )( '&8%,)*&
 ! !"
#
$%&'( )* +,"! -(, .+%(/$!$,
Proposição 1.7 !"# V $% !&'#()* +!,*-.#/0 1!%*&
20 3#-# 4$#/4$!- λ ∈ R, λ0 = 0.
50 3#-# 4$#/4$!- u ∈ V, 0u = 0.
60 ! λu = 0 !7,8#* λ = 0 *$ u = 0.
90 3#-# 4$#.&4$!- λ ∈ R ! u ∈ V, (−λ)u = λ(−u) = −(λu).
:0 3#-# 4$#/4$!- u ∈ V, −(−u) = u.
;0 ! u + w = v + w !7,8#* u = v.
<0 ! u, v ∈ V !7,8#* !=.&,! $% >$7.)* w ∈ V ,#/ 4$! u + w = v.
Prova:
 ! "#$%& λ0 = λ(0 + 0) = λ0 + λ0 '#()& '*%'*+#,),#& !" # !#!
-.+(+/)0,% )& '*%'*+#,),#& !$ ) !% # ) 0%.)123)% ,) %4&#*5)123)%
 !67 %4.#$%& 0 = λ0 + (−(λ0)) = (λ0 + λ0) + (−(λ0)) = λ0 + (λ0 +
(−(λ0))) = λ0 + 0 = λ0, +&.% 8# λ0 = 0.
9! "#$%& 0u = (0 + 0)u = 0u + 0u, '#() '*%'*+#,),# !&! -.+(+/)0,%
)& '*%'*+#,),#& !$ ) !% # ) 0%.)123)% ,) %4&#*5)123)% !67 %4.#$%&
0 = 0u + (−(0u)) = (0u + 0u) + (−(0u)) = 0u + (0u + (−(0u)) =
0u + 0 = 0u, +&.% 8#7 0u = 0.
:! ;# λ 6= 0 #0.3)% '#()& '*%'*+#,),#& !' # !( # '#(% +.#$ ,#&.)
'*%'%&+123)%7 u = 1u = (λ−1λ)u = λ−1(λu) = λ−10 = 0.
6! -.+(+/)0,% ) '*%'*+#,),# !& # % +.#$ 9 ,#&.) '*%'%&+123)%7 %4.#$%&
λu + (−λ)u = (λ + (−λ))u = 0u = 0. <#() %4&#*5)123)% !67 −(λu) =
(−λ)u. =0)(%>)$#0.#7 ?.+(+/)0,%@&# ) '*%'*+#,),# !#7 $%&.*)@&#
A?# −(λu) = λ(−u).
 !"! #$#%&
'
(&()* !
" #$%&' (%) %*+$%) $,)*-+'(%) ., (,/0'(' 1%2% ,0,$1.31/%4
Ex. Resolvido 1.8 !"# V $% !&'#()* +!,*-.#/0 1*&,-! 2$! &! V 6= {0}
!3,4#* V ,!% .353.,*& !/!%!3,*&0
Resolução: 5%+, 6*, ), ,71%7+$'$2%) *2' 8*791:'% f : R → V 6*, ),;' /7;,<
+%$' ,7+:'% V +,$.' /7=7/+%) ,-,2,7+%)> #%/) #'$' 1'(' λ ∈ R 1%$$,)#%7(,$.'
*2 ,-,2,7+% (/)+/7+% f(λ) (, V.
?%2, v ∈ V, v 6= 0. @,=7' f : R → V #%$ f(λ) = λv. A'$' 2%)+$'$
6*, f ., /7;,+%$'> +%2,2%) λ, µ ∈ R +'/) 6*, f(λ) = f(µ). @,&,2%) 2%)+$'$
6*, λ = µ. B%2% λv = f(λ) = f(µ) = µv, %C+,2%) λv − (µv) = 0. A,-%
/+,2 D (' #$%#%)/91:'% 4E +,2%) 0 = λv − (µv) = λv + (−µ)v = (λ − µ)v.
B%2% v 6= 0, #,-% /+,2 F (' 2,)2' #$%#%)/91:'%> ),G*, 6*, λ − µ = 0, /)+%
.,> λ = µ. ¤
1.3 Exerćıcios
Ex. 1.9 6!-.52$! &! !% )#7# $% 7*& .,!3& * )*3"$3,* V )*% #&
*'!-#()4*!& .37.)#7#& 8! $% !&'#()* +!,*-.#/ &*9-! R.
 4 V = R3, (x1, y1, z1)+(x2, y2, z2) = (x1+x2, y1+y2, z1+z2); α(x, y,z) =
(αx, αy, αz).
H4 V =
{(
a −b
b a
)
;a, b ∈ R
}
, %#,$'91:%,) *)*'/) (, M2.
F4 V =
{
(x, y) ∈ R2; 3x − 2y = 0
}
, %#,$'91:%,) *)*'/) (, R2.
D4 V = {f : R → R; f(−x) = f(x), ∀x ∈ R}, %#,$'91:%,) *)*'/) (, 8*791:%,)4
!4 V = P(R) = { #%-/7I%2/%) 1%2 1%,=1/,7+,) $,'/) } , %#,$'91:%,) *)*'/)
(, 8*791:%,)4
 ! !"
#
$%&'( )* +,"! -(, .+%(/$!$,
!" V = R2, (x1, y1)+(x2, y2) = (2x1−2y1, y1−x1), α(x, y) = (3αx, −αx.)
#" V = R2, (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2), α(x, y) = (αx, 0).
$" V =
{
(x, y, z,w) ∈ R4; y = x, z = w2
}
, %&'()*+,%'- .-.)/- 0' R4.
1" V = R × R∗, (x1, y1) + (x2, y2) = (x1 + x2, y1y2), α(x, y) = (αx, yα),
%20' R
∗ = R \ {0}.
Ex. 1.10 !"#$%! & '!#(%)*"&+,-&( '& ."(.()$+,-&( /010
Caṕıtulo 2
Subespaços Vetoriais
2.1 Introdução e Exemplos
 
 !"#$ %&'&$ ()$ *&+#,#,&-)$ .)- .&,")$ $ /.)(0 (")$ *& - &$+#1.)
%&"),!#2 3 & +)$$ &- # +,)+,!&*#*& *& 3 & # $)-# *& *)!$ *& $& $
&2&-&(")$ 4& - &2&-&(") *) +,4)+,!) $ /.)(0 (") /&- .)-) 3 #(*) - 25
"!+2!.#-)$ - &2&-&(") *) $ /.)(0 (") +), - &$.#2#,6 ) ,&$ 2"#*) .)(5
"!( # +&,"&(.&(*) #) $ /.)(0 (")7
Definição 2.1 !"# V $% !&'#()* +!,*-.#/0 1.2!%*& 3$! W ⊂ V 4! $%
&$5!&'#()* +!,*-.#/ 6! V &! 7*-!% &#,.&7!.,#& #& &!8$.9,!& )*96.():*!&;
 !"#$ 0 ∈ W;
 !"%$ ! u, v ∈ W !9,:#* u + v ∈ W;
 !"&$ ! u ∈ W !9,:#* λu ∈ W '#-# ,*6* λ ∈ R.
Observação 2.2 <*,! 3$! ,*6* &$5!&'#()* +!,*-.#/ W 6! $% !&'#()*
+!,*-.#/ V 4! !/! '-4*'-.* $% !&'#()* +!,*-.#/0 =& '-*'-.!6#6!& )*%$,#>
,.+#? #&&*).#,.+#? 6.&,-.5$,.+#& ! '"( &:#* @!-6#6#& 6* '-4*'-.* !&'#()*
+!,*-.#/ V. A !/!%!9,* 9!$,-* 6# #6.():#* 4! $% !/!%!9,* 6! W '*-
89
 ! !"
#
$%&'( )* +&,-+"! .(+ /-%(0$!$+
 !" !"#$%&'#(') *' u ∈ W '#(+$, −u = (−1)u ∈ W -'%, "('& . /$
-0,-,*"12+$, 3 4 ' -,0 !# 
Observação 2.3 567"$&'#(' {0} ' V *+$, *86'*-$12,* 7'(,0"$"* /, '*-$9
12, 7'(,0"$% V. :+$, 2;$&$/,* /' *86'*-$12,* 7'(,0"$"* (0"7"$"* 
Observação 2.4 <,(' =8' W >' *86'*-$12, 7'(,0"$% /' V *' ' *,&'#('
*' *+$, 7>$%"/$* $* *'?8"#('* 2,#/"12+,'*@
$ !"%& 0 ∈ W;
$ !'%& :' u, v ∈ W ' λ ∈ R '#(+$, u + λv ∈ W.
"#$%&'( %)*+,( '+-.'( #/#&0)'(1
Exemplo 2.5 :'A$ P∗n ⊂ Pn, /$/, -,0 P∗n = {p(x) ∈ Pn;p(0) = 0}.
"#.234+#&'( 4+# P∗n 5#6 7# 8%-'6 +& (+9#(0%:;' <#-'.2%) 7# Pn.
 = > 0')2,?'&2' ,+)' (# %,+)% #& x = 0, )'*'6 0#.-#,;# % P∗n.
@= A# p(x), q(x) ∈ P∗n #,-B%' p(0) + q(0) = 0 #6 0'.-%,-'6 p(x) + q(x) ∈
P∗n.
C= A# p(x) ∈ P∗n #,-B%' λp(0) = 0 0%.% 4+%)4+#. λ ∈ R. D((2&6 λp(x) ∈
P∗n.
Exemplo 2.6 B'0"C=8'&,* =8' S = {(x, y, z) ∈ R3; x + y + z = 0} >' 8&
*86'*-$12, 7'(,0"$% /' R
3.
 =
5
E ;)%.' 4+# (0, 0, 0) (%-2(8%F 0 + 0 + 0 = 0.
@= A# (x, y, z), (u, v, w) ∈ S #,-B%' (x + u) + (y + v) + (z + w) = (x + y +
z) + (u + v + w) = 0 #6 0'.-%,-'6 (x, y, z) + (u, v, w) ∈ S.
 !"! #$%&'()*+
,
-' . ./.012'3 !
"# $% (x, y, z) ∈ S %&'()* λx + λy + λz = λ(x + y + z) = 0 +),) -.)/-.%,
λ ∈ R. 011234 λ(x, y, z) ∈ S.
Exemplo 2.7 !"#$%&'& ! #&()$"*& +!",)"*! S = {y ∈ C2(R; R); y ′′ −
y = 0} !"%& y ′′ '&-'&#&"*. . %&'$/.%. %& #&()"%. !'%&0 %& y. 1&'$23
4)&0!# 4)& S 5& )0 #)6&#-.7+! /&*!'$.8 %& C2(R; R).
 # 5/),)3%&'% ) 6.&78()* &./) 1)'216)9 0 ′′ − 0 = 0;
:# $% y1, y2 ∈ S %&'()* (y1+y2) ′′−(y1+y2) = (y ′′1 −y1)+(y ′′2 −y2) = 0.
;*<*4 y1 + y2 ∈ S.
"# $% y ∈ S % λ ∈ R %&'()* (λy) ′′ − λy = λ(y ′′ − y) = 0. =*,')&'*4
λy ∈ S.
>%2?)3*1 8*3* %?%,8@A82* ) B%,2C8)78()* D% -.% *1 1%<.2&'%1 %?%3+/*1
1()* 1.E%1+)78*1 B%'*,2)21 D*1 ,%1+%8'2B*1 %1+)78*1 B%'*,2)21#
Exemplo 2.8 9&,.0 a1, . . . , an ∈ R & S = {(x1, . . . , xn) ∈ Rn; a1x1+· · ·+
anxn = 0}. :!#*'& 4)& S 5& )0 #)6&#-.7+! /&*!'$.8 %& R
n.
Exemplo 2.9 ; +!",)"*! %.# <)"7+=!&# +!"*5>").# %. '&*. ". '&*.? %&3
"!*.%! -!' C(R; R)? 5& )0 #)6&#-.7+! /&*!'$.8 %& F (R; R).
Exemplo 2.10 ; +!",)"*! %.# <)"7+=!&# f ∈ C([a, b]; R) *.$# 4)&
∫b
a
f(x)dx = 0
5& )0 #)6&#-.7+! /&*!'$.8 %& C([a, b]; R).
Exemplo 2.11 ; +!",)"*! %.# 3)',29%1 123@%',28)1 4).%'.%.# %& !'%&0
n +!0 +!&2+$&"*&# '&.$# 5& )0 #)6&#-.7+! /&*!'$.8 %& Mn(R).
Exemplo 2.12 9&,.0 m,n ∈ N +!0 m ≤ n. @"*=.! Pm 5& )0 #)3
6&#-.7+! %& Pn.
 ! !"
#
$%&'( )* +&,-+"! .(+ /-%(0$!$+
2.2 Interseção e Soma de Subespaços
Proposição 2.13 (Interseção de subespaços) !"#$ U ! W %&'!%(#)
*+,% -!.,/0#0% 1! V. 23.4#, U ∩ W 5! %&'!%(#*+, -!.,/0#6 1! V.
Prova:
"# $%&% 0 ∈ U ' 0 ∈ W '()*+% 0 ∈ U ∩ W;
 # ,' x, y ∈ U ∩ W ' λ ∈ R '()*+% x + λy ∈ U ' x + λy ∈ W. -%.)+()%/
x + λy ∈ U ∩ W.
Questão: $%& + (%)+01*+% 2+ 3.%3%4501*+% +15&+/ 3%2'&%4 +6.&+. 78'
U ∪ W 9' 48:'43+01% ;')%.5+< 2' V?
Resposta : =*+%# >+4)+ 1%(452'.+. V = R2, U = {(x, y) ∈ R2; x + y = 0}
' W = {(x, y) ∈ R2; x − y = 0}. =%)' 78' (1,−1) ∈ U ⊂ U ∪ W ' (1, 1) ∈
W ⊂ U ∪ W &+4 (1,−1) + (1, 1) = (2, 0) 6∈ U ∪ W.
,' U ' W 4*+% 48:'43+01%4 ;')%.5+54 2' 8& '43+01% ;')%.5+< V ' V ′ 9' 8&
48:'43+01% 2' V 78' 1%()'(?+ U ' W, 54)% 9'/ U ∪ W ⊂ V ′ '()*+% V ′ )'.9+
78' 1%()'. )%2%4 %4 ;')%.'4 2+ @%.&+ u + w, u ∈ U ' w ∈ W. A4)% &%)5;+
+ 4'B85()'
Definição 2.14 !"#$ U ! W %&'!%(#*+,% -!.,/0#0% 1! &$ !%(#*+, -!)
.,/0#6 V. 7!830$,% # %,$# 1! U ! W +,$, U+W = {u+w;u ∈ U,w ∈
W}.
Proposição 2.15 (Soma de subespaços) !"#$ U,W ! V +,$, 3#
1!830*+4#, #+0$#9 23.4#, U + W 5! &$ %&'!%(#*+, -!.,/0#6 1! V. :65!$
1, $#0%; U ∪ W ⊂ U + W.
Prova: C'.5678'&%4 78' U + W 9' 48:'43+01% ;')%.5+< 2' V.
"# $%&% 0 ∈ U ' 0 ∈ W '()*+% 0 = 0 + 0 ∈ U + W;
 ! ! "#$%&'%()
*
+, % ',-+ .% '/0%'1+(),' !
 " #$%&' x1, x2 ∈ U + W $()*&+ xj = uj + wj, uj ∈ U, wj ∈ W, j = 1, 2.
,-+.&/ 0$ λ ∈ R $()*&+ x1 + λx2 = u1 + w1 + λ(u2 + w2) = (u1 +
λu2) + (w1 + λw2) ∈ U + W, 1+20 U $ W 0*&+ 034$01&56+0 7$)+.2&20"
8+0).$'+0 93$ U ∪ W ⊂ U + W. #$%& v ∈ U ∪ W. #$ v ∈ U $()*&+
v = v + 0 ∈ U + W. #$ v ∈ W $()*&+ v = 0 + v ∈ U + W. :3 0$%&/
U ∪ W ⊂ U + W.
,2(;& 30&(;+ & (+)&56*&+ &62'&/ 031+(<& 93$ V ′ 0$%& 3' 034$01&56+
;$ V 93$ 6+()$(<& U $ W. =$0)$ 6&0+/ 1&.& )+;+ u ∈ U ⊂ V ′ $ )+;+
w ∈ W ⊂ V ′ )$'+0 u + w ∈ V ′, +3 0$%&/ U + W ⊂ V ′. >0)& +40$.7&56*&+
(+0 1$.'2)$ .$-20).&. & 0$-32()$
Proposição 2.16 !"#$ V %$ !&'#()* +!,*-.#/ ! U ! W &%0!&'#()*&
+!,*-.#.& 1! V. 23,4#* U + W 5! * $!3*- &%0!&'#()* +!,*-.#/ 1! V 6%!
)*3,5!$ U ∪ W. 2$ *%,-#& '#/#+-#&7 &! V ′ 5! %$ &%0!&'#()* +!,*-.#/ 1!
V 6%! )*3,5!$ U ∪ W !3,4#* U ∪ W ⊂ U + W ⊂ V ′.
Definição 2.17 !"#$ U ! W &%0!&'#()*& +!,*-.#.& 1! %$ !&'#()* +!8
,*-.#/ V. 9.:!$*& 6%! U+W 5! # &*$# 1.-!,# 1! U ! W &! U∩W = {0}.
;!&,! )#&* %&#-!$*& # 3*,#()4#* U ⊕ W '#-# -!'-!&!3,#- U + W.
Observação 2.18 ;*,! 6%! ,-.+.#/$!3,! {0} ⊂ U ∩ W &! U ! W &4#*
&%0!&'#()*& +!,*-.#.&<
Proposição 2.19 (Soma direta de subespaços vetoriais) !"#$ U !
W &%0!&'#()*& +!,*-.#.& 1! %$ !&'#()* +!,*-.#/ V. =!$*& V = U⊕W &!
! &*$!3,! &! '#-# )#1# v ∈ V !>.&,.-!$ %$ 5%3.)* u ∈ U ! %$ 5%3.)*
w ∈ W &#,.&?#:!31* v = u + w.
Prova: #31+(<& 93$ V = U ⊕ W, 20)+ ?$/ V = U + W $ U ∩ W = {0}.
>()*&+/ ;&;+ v ∈ V $@20)$' u ∈ U $ w ∈ W 0&)20A&B$(;+ v = u + w.
C3$.$'+0 '+0).&. 93$ )&D 1!)*$'*&.()4#* ?$ ?3(26&" #31+(<& 93$ $@20)&'
u ′ ∈ U $ w ′ ∈ W )&20 93$ v = u ′ + w ′. >()*&+/ u + w = u ′ + w ′, + 93$
 !"
#
$%&'( )* +&,-+"! .(+ /-%(0$!$+
!"#$!%& '" u − u ′ = w ′ − w. (&) u − u ′ ∈ U ' w ′ − w ∈ W '* #+,-&.-+*
u − u ′ = w ′ − w ∈ U ∩ W = {0}, +/ )'0& u = u ′ ' w = w ′.
1/#+.2& &3+,& 4/' #&,& %&5& v ∈ V '6!)-&" /" 7/.!%+ u ∈ U ' /"
7/.!%+ w ∈ W )&-!)8&9'.5+ v = u + w. 7: %$&,+ 4/' V = U + W. ;')-&
"+)-,&, 4/' U ∩ W = {0}. <=>!&"'.-'* 0 ∈ U ∩ W. 1'0& v ∈ U ∩ W, !)-+
7'* v ∈ U ' v ∈ W. :.-?&+* '6!)-'" /" 7/.!%+ u ∈ U ' /" 7/.!%+ w ∈ W
)&-!)8&9'.5+ v = u + w. <=)',>' 4/' v = u + w = (u + v) + (w − v) %+"
u + v ∈ U ' w − v ∈ W '* #'$& /.!%!5&5' 5& 5'%+"#+)!@%?&+* 5'>'"+) -',
u = u + v ' w = w − v, !)-+ 7'* v = 0. A+3+* U ∩ W = {0}.
B$-',.&-!>&"'.-'* #+5',7C&"+) )/#+, & '6!)-D'.%!& 5' v 6= 0 '" U ∩ W
' 5&7C +=-',7C&"+) v = 2v − v = 4v − 3v, 5/&) 5'%+"#+)!@%?+') 5!)-!.-&) #&,&
v 07& 4/' 2v, 4v ∈ U, 2v 6= 4v ' −v,−3v ∈ W.
Exemplo 2.20 !"#$%&! %&! R3 '! ( )*+( ,#"!-( ,! U = {(x, y, z) ∈
R
3; x + y + z = 0} ! W = {(x, y, z) ∈ R3; x = y = 0}.
E+-' 4/' W 7' 5' 8&-+ /" )/=')#&@%+ >'-+,!&$ 5' R3 #+!) W ={(x, y, z) ∈
R
3; x = 0}∩{(x, y, z) ∈ R3; y = 0} +/* &$-',.&-!>&"'.-'* )' u1 = (x1, y1, z1),
u2 = (x2, y2, z2) ∈ W '.-?&+ x1 = y1 = x2 = y2 = 0 ' u1+u2 = (0, 0, z1+z2)
7' %$&,&"'.-' /" '$'"'.-+ 5' W.
1' λ ∈ R '.-?&+
λu1 = λ(0, 0, z1) = (λ0, λ0, λz1) = (0, 0, λz1) ∈ W.
F!.&$"'.-'* (0, 0, 0) ∈ W, + 4/' %+.%$/! & #,+>& 5' 4/' W 7' /" )/G
=')#&@%+ >'-+,!&$H
I,+))'3/!.5+* 5&5+ (x, y, z) ∈ R3 #+5'"+) ')%,'>',
(x, y, z) = (x, y,−x − y) + (0, 0, z + x + y)
' %+"+ (x, y, −x − y) ∈ U ' (0, 0, z + x + y) ∈ W +=-'"+) R3 = U + W.
 ! ! "#$%&'%()
*
+, % ',-+ .% '/0%'1+(),' !
"#$%& &'()& *($%)&) +,# U ∩ W = {0}. -#.& (x, y, z) ∈ U ∩ W. /#*($



x + y + z = 0
x = 0
y = 0
⇐⇒ (x, y, z) = (0, 0, 0).
Ex. Resolvido 2.21 !"#$%&'& !# #()&#*+,-!# %& R3 %+%!# *!'
U = {(x, y, z) ∈ R3; x = 0} & V = {(x, y, z) ∈ R3; y = 0}.
.!#/'& 0(& R
3 = U + V, 1+# + #!1+ "2+! 3& %$'&/+4
Resolução: 0&1( (x, y, z) ∈ R3 2(1#*($ #$3)#4#)
(x, y, z) = (0, y, z) + (x, 0, 0) ∈ U + V,
2(5$ (0, y, z) ∈ U # (x, 0, 0) ∈ V. 6()%&7%(8 R3 = U + V.
9( #7%&7%(8 & $(*& 7:&( ;# 15)#%& 2(5$ U ∩ V 6= {(0, 0, 0)}, 2(5$8 2()
#<#*2=(8 (0, 0, 1) ∈ U ∩ V. ¤
Definição 2.22 5&6+1 U1, . . . , Un !"# $%&'( )#*(+,%, -# !. # $%&'(
)#*(+,%/ V. 0 (.% -# U1 % Un 1# -#23,-% $(+
U1 + · · · + Un = {u1 + · · · + un;uj ∈ Uj, j = 1, . . . , n}.
Definição 2.23 4#5%. U1, . . . , Un !"# $%&'( )#*(+,%, -# !. # $%&'(
)#*(+,%/ V. 6,7#.( 8!# % (.% -# U1 % Un 1# !.% (.% -,+#*% #
Uj ∩
(
U1 + · · · + Ûj + · · · + Un
)
= {0}, j = 1, . . . n,
#. 8!# ( *#+.( Ûj -#)# #+ (.,*,-( -% (.%9 :# *# '% ( ! %+#.( 
% 3(*%&';%( U1 ⊕ · · · ⊕ Un $%+% -#3(*%+ % (.% -# U1 % Un.
 ! !"
#
$%&'( )* +&,-+"! .(+ /-%(0$!$+
Observação 2.24 ! "#$%" &'(
0 ∈ Uj ∩
(
U1 + · · · + Ûj + · · · + Un
)
)( U1, . . . , Un )*+" )'#(),+-.") $(/"0%+%)1
Proposição 2.25 2(3+4 U1, . . . , Un )'#(),+-.") $(/"0%+%) 5( '4 (),+-."
$(/"0%+6 V. !7/*+" V = U1 ⊕ · · · ⊕ Un )( ( )"4(7/( )( ,+0+ .+5+ v ∈ V
(8%)/(9 ,+0+ .+5+ j = 1, . . . , n, '4 '7%." uj ∈ Uj /+6 &'( v = u1+· · ·+un.
Prova: " #$%&' () '*('+%,' -' .' #$%#%/0123'% 4564
Exemplo 2.26 :")/0( &'( P2 ( )"4+ 5%0(/+ 5") )(;'%7/() )'#(),+-.")
$(/"0%+%) U1 = {a0; a0 ∈ R}, U2 = {a1x;a1 ∈ R} ( U3 = {a2x2;a2 ∈ R}.
7'.% p(x) ∈ P2, 8)9%/ p(x) = a0 + a1x + a2x2, #'$' 2)$8%/ 2%):20)*8)/
a0, a1, a2 ∈ R. "//09; P2 = U1 + U2 + U3.
<)$0:=>)9%/ =>) ' /%9' () .0$)8'4
54 ?%/8$)9%/ =>) U1 ∩ (U2 + U3) = {0}. @)A' p(x) ∈ U1 ∩ (U2 + U3).
B*83'% )C0/8)9 a0, a1, a2 ∈ R 8'0/ =>) p(x) = a0 = a1x + a2x2. @)
p(x) *3'% D%//) % #%+0*E%90% *>+% 8)$(F'9%/ >9 #%+0*E%90% .) ,$'> 0,
a0, 2%0*20.0*.% 2%9 >9 .) ,$'> *% 9(F*09% 5; a1x + a2x
2, % =>) ()
>9 'G/>$.%4 H%,%; p(x) = 0.
 4 ?%/8$)9%/ =>) U2 ∩ (U1 + U3) = {0}. @)A' p(x) ∈ U2 ∩ (U1 + U3).
B*83'% )C0/8)9 a0, a1, a2 ∈ R 8'0/ =>) p(x) = a1x = a0 + a2x2. @)
p(x) *3'% D%//) % #%+0*E%90% *>+% 8)$(F'9%/ >9 #%+0*E%90% .) ,$'> 5;
a1x, 2%0*20.0*.% 2%9 >9 .) ,$'> I J2'/% a2 = 0K %> ; a0 + a2x
2,
J2'/% a2 6= 0K; % =>) () >9 'G/>$.%4 H%,%; p(x) = 0.
L4 ?%/8$)9%/ =>) U3 ∩ (U1 + U2) = {0}. @)A' p(x) ∈ U3 ∩ (U1 + U2).
B*83'% )C0/8)9 a0, a1, a2 ∈ R 8'0/ =>) p(x) = a2x2 = a0 + a1x. @)
p(x) *3'% D%//) % #%+0*E%90% *>+% 8)$(F'9%/ >9 #%+0*E%90% .) ,$'> ;
a2x
2, 2%0*20.0*.% 2%9 >9 .) ,$'> I J2'/% a1 = 0K %> 5; a0 + a1x,
J2'/% a1 6= 0K; % =>) () >9 'G/>$.%4 H%,%; p(x) = 0.
 !"! #$#%&
'
(&()* !
2.3 Exerćıcios
Ex. 2.27 !"#$%&! '! !( )*+* &( +,' #-!.' */*#0, , '&/),.1&.-,
W 2! &( '&/!'3*4), 5!-,"#*6 +, !'3*4), 5!-,"#*6 V. 7*', .8*, '!1*(
!'3!)#$)*+*'9 ),.'#+!"! *' ,3!"*4)8,!' &'&*#':
"# V = M2, W =
{(
a b
−a c
)
;a, b, c,∈ R
}
.
 # V = R4, W = {(x, x, y, y); x, y ∈ R} .
$# V = Pn(R), W = {p ∈ Pn(R); p(0) = p(1)} .
%# V = Mn, &'&' B ∈ Mn, &()*' W = {A ∈ Mn; BA = 0} .
!# V = Rn, W = {(x1, x2, · · · , xn); a1x1 + · · · + anxn = 0} , +*&( a1, . . . ,
an ∈ R ,-'+ &'&+,#
.# V = Mn×1, W = {X ∈ Mn×1;AX = 0} , +*&( A ∈ Mm×n /( &'&'#
0# V = Pn(R), W = {p ∈ Pn(R);p′(t) = 0, ∀t ∈ R} .
1# V = Mn, W = {A ∈ Mn; At = A} .
2# V = Mn,W = {A ∈ Mn;At = −A} .
"3# V = C∞(R; R),W = {f ∈ C∞(R; R); 456x→+∞ f(x) = 0} .
""# V = F (R; R),W = {f ∈ F (R; R); f(x0) = 0} , x0 ∈ R.
Ex. 2.28 ;#<*9 !( )*+* &( +,' #-!.' */*#0,9 '! * *$"(*4)8*, 2! 5!"=
+*+!#"* ,& >*6'*9 1&'-#$)*.+, '&* "!'3,'-*: #'-, 2!9 3",5*.+, '! >,"
5!"+*+!#"* ,& +*.+, &( ),.-"*=!0!(36, '! >," >*6'*:
"# 7( W1 ( W2 ,-'+ ,8,9(,:';<+, &( 86 (,:';<+ =(>+?5'4 V (*>-'+ W1∪W2
/( ,89(,:';<+ &( V.
 ! !"
#
$%&'( )* +&,-+"! .(+ /-%(0$!$+
 " #$%&' W1 $ W2 ()*$(+&,-.( /$ )' $(+&,-. 0$1.23&4 V. 5617&. W1∪W2 8$
()*$(+&,-. /$ V ($9 $ (.'$61$ ($9 W1 ⊆ W2 .) W2 ⊆ W1. :#);$(17&.<
'.(12$ =)$ ($ W 8$ ()*$(+&,-. /$ V $ x0, y0 ∈ V (7&. 1&3( =)$ x0 ∈ W
$ y0 6∈ W $617&. x0 + y0 /∈ W $ )($>."?
Ex. 2.29 ! "#$# %&'! #(#%)* '+"*+&,#, *- -.('-/#0"*- U+W ' U∩W1
*+$' U, W -2#* -.('-/#0"*- $* '-/#0"* 3'&*,%#4 V %+$%"#$*5
@" U =
{
(x, y) ∈ R2;y = 0
}
, W =
{
(x, y) ∈ R2; x = 2y
}
, V = R2.
 " U =
{(
a 0
0 b
)
;a, b ∈ R
}
, W =
{(
0 c
0 d
)
; c, d ∈ R
}
, V =
M2.
A" V = P3(R), U = {p(t) ∈ V ;p′′(t) = 0} , W = {q(t) ∈ V ; q′(t) = 0} .
Ex. 2.30 6',%78.'1 '! "#$# .! $*- %&'+- #(#%)*1 -' V = U ⊕ W.
95
V = R2, U =
{
(x, y) ∈ R2; 2x + 3y = 0
}
,
W =
{
(x, y) ∈ R2; x − y = 0
}
.
:5 V = M3, U =





a b 0
0 0 c
0 0 d

 ; a, b, c, d ∈ R



,
W =





0 0 e
f g 0
h i 0

 ; e, f, g, h, i ∈ R



.
;5 V = P3(R), U = {p(t) ∈ P3(R); p(1) = p(0) = 0} ,
W = {q(t) ∈ P3(R); q′(t) = 0,∀t ∈ R} .
 !"! #$#%&
'
(&()* !
Ex. 2.31 ! "#$# %! $&' ()*+' #,#(-&. $#$& U '%,*'/#0"& $* V. *+1
"&+)2#2 & '%,*'/#0"& '%/3*!*+)#2 $* U. (')& 4*. & '%,*'/#0"& W $* V
)#3 5%* V = U ⊕ W.
"# V = R3, U = {(x, y, 0); x, y ∈ R} .
 # V = P3(R), U = {p(t) ∈ P3(R); p ′′(t) = 0,∀t ∈ R} .
$# V = M3, U = {A ∈ M3;At = A} .
%# V = M2×1, U = {X ∈ M2×1;AX = 0} , &'() A =
(
1 1
0 1
)
.
 ! !"
#
$%&'( )* +&,-+"! .(+ /-%(0$!$+
Caṕıtulo 3
Combinações Lineares
3.1 Introdução e Exemplos
 
 !"# $" %&'()*+," &$*-. ". /+- +! #+0-#'&1%" 2-*". &, (- +! #+0%"$3
4+$*" 5- +! -#'&1%" 2-*". &, /+- (- 6-%7&5" %"! .-,&1%8&" 9& &5 1%8&" 5-
2-*".-# - *&!0(-! %"! .-,&1%8&" 9& !+,* ', %&1%8&" '". -#%&,&.: ;! "+*.&# '&3
,&2.&#< /+&$5" #"!&!"# 5" # 2-*".-# 5- +! #+0-#'&1%" 2-*". &, "+ !+,* 3
', %&!"# +! 2-*". 5" #+0-#'&1%" '". +! -#%&,&.< " .-#+,*&5" (- +! -,-!-$*"
5-#*- #+0-#'&1%": =+&$5" %"!0 $&!"# .-'-* 5&# 2->-# -#*&# &1%8"-# *-!"#
" /+- %7&!&!"# 5- %"!0 $&1%8&" , $-&. -$*.- 2-*".-#: ?& # '.-% #&!-$*-<
Definição 3.1 !"#$ u1, . . . , un !%!$!&'() *! +$ !),#-.( /!'(01#% V.
213!$() 4+! u 5! %"!0 $&1%8&" , $-&. *! u1, . . . , un )! !61)'10!$ &5+$!0()
0!#1) α1, . . . , αn '#1) 4+! u = α1u1 + · · · + αnun
Observação 3.2 !"#$ U +$ !),#-.( /!'(01#% ! V ⊂ U +$ )+7!),#-.(
/!'(01#%8 ! u1, . . . , un ∈ V ! α1, . . . , αn ∈ R !&'9#( # .($71&#-.9#( %1&!#0
α1u1 + · · · + αnun ,!0'!&.! # V.
Exemplo 3.3 :$ P2, ( ,(%1&;($1( p(x) = 2 + x
2
5! +$# .($71&#-.9#(
*() ,(%1&;($1() p1(x) = 1, p2(x) = x ! p3(x) = x
2.
@A
 ! !"
#
$%&'( )* (+,$-! .
/
(01 '$-0!201
"#$%# &'( )*' p(x) = 2p1(x) + 0p2(x) + p3(x).
Exemplo 3.4 !"#$%&! %&! !' P2, ( )(*#+,('#( p(x) = 1 + x
2
-! &'.
/('0#+.1/2.( 3(4 )(*#+,('#(4 q1(x) = 1, q2(x) = 1+x ! q3(x) = 1+x+x
2.
+(',-$#./$ '0,/0%(#( 01*.'(/$ ('#-$ α,β ' γ %#-$ )*' p(x) = αq1(x) +
βq2(x) + γq3(x). 2* $'3#4 5(',-$#./$ '0,/0%(#( α,β ' γ $#%-$6#7'08/
1 + x2 = α + β(1 + x) + γ(1 + x + x2) = α + β + γ + (β + γ)x + γx2,
)*' 1' ')*-&#9'0%' #/ $-$%'.#



α + β + γ = 1
β + γ = 0
γ = 1
⇐⇒ α = 1, β = −1 ' γ = 1.
3.2 Geradores
Definição 3.5 5!6.' V &' !4).1/( 7!8("#.* ! S &' 4&0/(+6&+8( +2.(
7.9#( 3! V. :4."!'(4 ( 4-;'0(*( [S] ).". 3!+(8." ( /(+6&+8( 3! 8(3.4 .4
/('0#+.1/2(!4 *#+!."!4 3(4 !*!'!+8(4 3! S. <' (&8".4 ).*.7".4= u ∈ [S]
4! !>#48#"!' α1, . . . , αn ∈ R ! u1, . . . , un ∈ S 8.#4 %&! u = α1u1 + · · · +
αnun.
Proposição 3.6 5!6.' V &' !4).1/( 7!8("#.* ! S &' 4&0/(+6&+8( +2.(
7.9#( 3! V. <+82.( [S] -! &' 4&0!4).1/( 7!8("#.* 3! V.
Prova:
:; </./ S 6= ∅ '=-$%' u ∈S. >/?/4 0 = 0u ∈ [S].
 !"! #$%&'(%$) !
"# $% u, v ∈ [S] %&'()* %+,-'%. α1, . . . , αn, β1, . . . , βm ∈ R % u1, . . . , un,
v1, . . . , vm ∈ S '),- /0% u = α1u1+· · ·+αnun % v = β1v1+· · ·+βmvm.
1--,.2 3)4) '*5* λ ∈ R, '%.*-
u + λv = α1u1 + · · · + αnun + λ(β1v1 + · · · + βmvm)
= α1u1 + · · · + αnun + λβ1v1 + · · · + λβmvm ∈ [S].
Definição 3.7 !"#$ S ! V %&$& #%'$#( )'*!$&+ ,-! [S] .! & +-/
0!+1#2%& 3!4&*'#5 6!*#7& 1&* S. 8+ !5!$!94&+ 7! S +:#& %;#$#7&+ 7!
6%4)5*4%- 7! [S]. ! S = {u1, . . . , un} 4#$0.!$ -+#*!$&+ # 9&4#2%:#&
[S] = [u1, . . . , un].
Proposição 3.8 !"#$ S ! T +-0%&9"-94&+ 9:#&/3#<'&+ 7! -$ !+1#2%&
3!4&*'#5 V. =!$&+
>( S ⊂ [S];
?( ! S ⊂ T !94:#& [S] ⊂ [T ];
@( [[S]] = [S];
A( ! S .! -$ +-0!+1#2%& 3!4&*'#5 !94:#& S = [S];
B( [S ∪ T ] = [S] + [T ].
Prova:
!# $% u ∈ S %&'()* u = 1u ∈ [S];
"# $% u ∈ [S] %&'()* %+,-'%. α1, . . . , αn ∈ R % u1, . . . , un ∈ S '),- /0%
u = α1u1+· · ·+αnun. 7*.* S ⊂ T '%.*- u1, . . . , un ∈ T %2 3*4')&'*2
u ∈ [T ];
 ! !"
#
$%&'( )* (+,$-! .
/
(01 '$-0!201
 " #$%& '($) * +$,(- ./&.&,'012-&3 [S] ⊂ [[S]]. 4$5- u ∈ [[S]]. 4$67$ +-
+$89'012-& :7$ u ;$ 7)- 1&)<'9-012-& %'9$-/ +$ $%$)$9(&, +$ [S], )-,
1&)& 1-+- $%$)$9(& +$ [S] ;$ 7)- 1&)<'9-012-& %'9$-/ +$ $%$)$9(&, +$
S /$,7%(- :7$ u ;$ 7)- 1&)<'9-012-& %'9$-/ +$ $%$)$9(&, +$ S, &7 ,$5-3
u ∈ [S];
=" #$%& '($) *3 S ⊂ [S]. 4$5- u ∈ [S]. >9(2-& u ;$ 7)- 1&)<'9-012-& %'9$-/
+$ $%$)$9(&, +$ S. ?&)& S ;$ 7) ,7<$,.-01& @$(&/'-%3 $,(- 1&)<'9-012-&
%'9$-/ ;$ 7) $%$)$9(& +$ S;
A" 4$5- u ∈ [S∪ T ]. #&/ +$89'012-&3 $B',($) α1, . . . , αn, β1, . . . , βm ∈ R $
u1, . . . , un ∈ S $ v1, . . . , vm ∈ T (-', :7$
u = α1u1 + · · · + αnun + β1v1 + · · · + βmvm
= (α1u1 + · · · + αnun) + (β1v1 + · · · + βmvm) ∈ [S] + [T ].
C$1'./&1-)$9($3 ,$ u ∈ [S] + [T ] $9(2-& u = v + w 1&) v ∈ [S] $ w ∈
[T ]. D$,,- E&/)-3 $B',($) α1, . . . , αp, β1, . . . , βq ∈ R $ v1, . . . , vp ∈ S
$ w1, . . . , wq ∈ T (-', :7$
u = v + w = α1v1 + · · · + αpvp + β1w1 + · · · + βqwq ∈ [S ∪ T ].
Definição 3.9 !"#$%& '(# ($ #&)*+,% -#.%/!*0 V 1# 89'(-)$9($ 6$/-+&
&# #2!&.!/ ($ &(3,%45(4.% 64!.% S ⊂ V .*0 '(# V = [S].
42-& $B$).%&, +$ $,.-01&, @$(&/'-', 89'(-)$9($ 6$/-+&,F
*" Pn(R) = [1, x, . . . , x
n];
!" R
n
;$ 6$/-+& .&/
e1 = (1, 0, . . . , 0), e2 = (0, 1, 0, . . . , 0), . . . , en = (0, . . . , 0, 1).
 !"! #$%&'(%$) 
 ! Mm×n "# $#%&'( )#*&+ ,&-%./#+ Ekl = (δ
(k,l)
i,j ), k = 1, . . . , m, l =
1, . . . n, (0'#
δ
(k,l)
i,j =
{
1 +# (i, j) = (k, l)
0 1&+( 1(0-%"&%.( .
Exemplo 3.10 !"# P(R) $ !%&#'($ )!*$+,#- .$+/#0$ &$+ *$0$% $% &$1
-,23$/,$%4 56+/#/$% 78! P(R) 29#$ :! 62,*#/!2*! ;!+#0$4
2(-# 34# Pn(R) ⊂ P(R) )&%& -('( n ∈ N. 5# P(R) 6(++# 70.-&,#0-#
$#%&'( #8.+-.%.&, )(*.09(,.(+ p1(x), . . . , pn(x) -&.+ 34#
P(R) = [p1(x), . . . , pn(x)].
5#:& N ( $%&4 ,&.+ &*-( '#0-%# (+ )(*.09(,.(+ p1(x), . . . , pn(x). "; #<.'#0-#
34# xN+1 0=&( )('# +#% #+1%.-( 1(,( 1(,>.0&?1=&( *.0#&% '# p1(x), . . . , pn(x)
#@ &++.,@ xN+1 6∈ [p1(x), . . . , pn(x)] = P(R). A,& 1(0-%&'.?1=&(!
2(-# 34# [1, x, x2, . . . ] = P(R).
Exemplo 3.11 !"# V 8/ !%&#'($ )!*$+,#- ;!+#0$ &$+ u1, . . . , un. <$%1
*+! 78! %!= &$+ !>!/&-$= u1 :! 8/# ($/?,2#'(9#$ -,2!#+ 0! u2, . . . , un
!2*9#$ V :! ;!+#0$ &$+ u2, . . . , un.
B#<#,(+ ,(+-%&% 34# 34&*34#% u ∈ V +# #+1%#<# 1(,( 4,& 1(,>.0&?1=&(
*.0#&% '# u2, . . . , un. 5&>#,(+ 34# #8.+-#, α1, . . . , αn ∈ R -&.+ 34# u =
α1u1+· · ·+αnun # #8.+-#, -&,>"#, β1, . . . , βn−1 +&-.+6&/#0'( u1 = β1u2+
· · · + βn−1un. C(,>.0&0'( #+-&+ .06(%,&?1=(#+@ (>-#,(+
u = α1(β1u2 + · · · + βn−1un) + α2u2 + · · · + αnun
= (α1β1 + α2)u2 + · · · + (α1βn−1 + αn)un ∈ [u2, . . . , un].
Exemplo 3.12 !"#/ U = {(x, y, z, t) ∈ R4; x − y + t + z = 0} ! V =
{(x, y, z, t) ∈ R4; x + y − t + z = 0}. @2($2*+! 8/ ($2"82*$ 62,*$ 0!
;!+#0$+!% &#+# $% %!;8,2*!% %8?!%&#'($% )!*$+,#,%A U, V, U∩V ! U+V.
 ! !"
#
$%&'( )* (+,$-! .
/
(01 '$-0!201
"# $% (x, y, z, t) ∈ U %&'()* y = x + z + t %+ ,*-')&'*+
(x, y, z, t) = (x, x+z+t, z, t) = x(1, 1, 0, 0)+z(0, 1, 1, 0)+t(0, 1, 0, 1),
./'* 0%+
U = [(1, 1, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1)].
1# $% (x, y, z, t) ∈ V %&'()* t = x + y + z %+ ,*-')&'*+
(x, y, z, t) = (x, y, z, x+y+z) = x(1, 0, 0, 1)+y(0, 1, 0, 1)+z(0, 0, 1, 1),
./'* 0%+
V = [(1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1)].
 # $% (x, y, z, t) ∈ U ∩ V %&'()*
{
x − y + t + z = 0
x + y − t + z = 0,
23% .4,5.6) %4 x = −z % y = t.
7%/'% 4*8*+ (x, y, z, t) = (x, y, −x, y) = x(1, 0, −1, 0) + y(0, 1, 0, 1)
%+ ,*-')&'*+
U ∩ V = [(1, 0, −1, 0), (0, 1, 0, 1)].
!# 9*4* U + V = [U] + [V ] = [U ∪ V ], '%4*/ 23%
U + V = [(1, 1, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1),
(1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1)]
= [(1, 1, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (1, 0, 0, 1), (0, 0, 1, 1)].
:;/%-<% 23%
(1, 1, 0, 0) = (1, 0, 0, 1) + (0, 1, 1, 0) − (0, 0, 1, 1)
%+ ,*-')&'*+
U + V = [(0, 1, 1, 0), (0, 1, 0, 1), (1, 0, 0, 1), (0, 0, 1, 1)].
=%-%4*/ 4)./ )8.)&'% 23% %/'% 0% * &034%-* 40>&.4* 8% ?%-)8*-%/ ,)-)
* /3;%/,)@6* U + V.
 ! ! "#"$%
&
'%'() !
3.3 Exerćıcios
Ex. 3.13 !"! #!$! %& $'( (%)#'*+%*,'( S ⊆ V- '*$. V /. ' .(0!1#'
2.,'"3!4 3*$3#!$'- .*#'*,"!" ' (%).(0!1#' 5."!$' 0'" S- 3(,' /.- [S].
"# S = {(1, 0), (2,−1)} , V = R2.
$# {(1, 1, 1), (2, 2, 0)} , V = R3.
 # S =
{
1, t, t2, 1 + t3
}
, V = P3(R).
%# S =
{(
0 1
0 0
)
,
(
0 0
−1 0
)}
, V = M2.
Ex. 3.14 6& #!$! %& $'( 3,.*( !)!37' .*#'*,"!" %& (%)#'*+%*,' S-
8*3,'- 9%. 5.". ' (%).(0!1#' 2.,'"3!4 W $' .(0!1#' 2.,'"3!4 V.
"# W =
{
(x, y, z) ∈ V .= R3; x − 2y = 0
}
.
$# W = {p ∈ V .= P3(R); p′(t) = 0,∀t ∈ R} .
 # W = {A ∈ V .= M2; At = A} .
%# W = {X ∈ V .= M3×1; AX = 0} , &'()
A =


0 1 0
2 1 0
1 1 4

 .
Ex. 3.15 6*#'*,"!"- .& #!$! %& $'( 3,.*( !)!37'- '( (%)#'*+%*,'( S
$' .(0!1#' 2.,'"3!4 V 9%. 5."!& U- W- U ∩ W . U + W.
"# U = [(1, 0, 0), (1, 1, 1)], W = [(0, 1, 0), (0, 0, 1)], V = R3.
$# U =
{
(x, y, z) ∈ R3; x + y = 0
}
, W = [(1, 3, 0), (0, 4, 6)], V = R3.
 ! !"
#
$%&'( )* (+,$-! .
/
(01 '$-0!201
 " U = {A ∈ M2;At = A} , W =
[(
1 1
0 1
)]
, V = M2.
#" U = [t3 + 4t2 − t + 3, t3 + 5t2 + 5, 3t3], W = [t3 + 4t2, t − 1, 1], V =
P3(R).
Ex. 3.16 !"#$%& ' ()!*'$+)$"' ,'-.&/' 0'- 1#"'-#( /' #(0&2*' 1#3
"'-4&5 P3(R) 6)# 7#-&. '( (#7)4$"#( ()!#(0&2*'(8
 ! U = {p ∈ P3(R); p(1) = p(0) = 0} ,
"! W = {p ∈ P3(R);p′′(t) = 0,∀t ∈ R} ,
#! U ∩ W.
Ex. 3.17 9'("-# 6)# 1, $%& 2x ∈ [ &'( 2x, $%&2 x].
Ex. 3.18 :#-4;6)# (# P2(R) <# 7#-&/' 0'- 1 + x, x + 2x
2
# 1 − x2.
Caṕıtulo 4
Dependência Linear
4.1 Introdução e Exemplos
 
 !"#$%&'( ")&*+, + " *-&'."+/ - - 0*+". +*- .* '/ *-#"1! 2*& +,"(
#+ !'+"/ - *)! )&+"+ '/ .*&*+/,)". ! )3')& .* 2*& +*- .* / . 
4'* 4'"(4'*+ 2*& + . *-#"1! */ 4'*-&5" #'.*--* -*+ *-!+,& ! / ! /6
7,)"1!5" (,)*"+ . - 2*& +*- .*-&* ! )3')& 8 9 + *:*/#( ; -* v * w 0*+"/
'/ *-#"1! V *)&5" #"+" 4'"(4'*+ u ∈ V $* # --$%2*( *)! )&+"+ *-!"("+*- α
* β -"&,-<"=*). u = αv + βw, ' -*3"
αv + βw − 1u = 0.
> &* 4'* " ! /7,)"1!5" (,)*"+ "!,/" $* )'("; */7 +" )*/ & . - - *-!"("+*-
4'* "#"+*!*/ )" -'" < +/"1!5" -5" )'( -8
?*3"/ - "0 +" " -*0',)&* -,&'"1!5" @ -*+$" # --$%2*( *)! )&+"+ *-!"("+*-
α,β * γ, )5" & . - )'( -; .* / . 4'*; */ R3 &*)A"/ -
α(1, 0, 0) + β(0, 1, 0) + γ(0, 0, 1) = (0, 0, 0)?
B +*-# -&" $*; 72,"/*)&* )5" 8 C-& -,0),D!" 4'* )5" $* # --$%2*( *-!+*2*+
)*)A'/ . - 2*& +*- "!,/" ! / ! /7,)"1!5" (,)*"+ . - '&+ - . ,-8 C-& 
! )&+"-&" ! / 4'* ! ++* ! / - 2*& +*- u, v * w . *:*/#( ")&*+, +8
EF
 ! !"
#
$%&'( )* +,",-+
.
,- $! '$-,!/
"#$ %&'() *&+(,-). )* /&()'&* -) 0',$&,') &1&$02) 3#4'-4$ #$4 %&'(4
-&0&+-5&+%,4 &+('& #$ & )#(') &+6#4+() 6#&. +) *&3#+-). )* ('5&* /&()'&*
*74) ,+-&0&+-&+(&*8
9&:4$)*. %)$ 4* -&;+,<%7)&* & &1&$02)* 6#& *&3#&$ %)$) 0)-&$)*
()'+4' &*(&* %)+%&,()* $4,* 0'&%,*)*8
Definição 4.1 !"#$%& '(# ($) &#'*(+#,-!) .# /#0%1#& u1, . . . , un .#
($ #&2)3-% /#0%1!)4 V 5# 4!,#)1$#,0# !,.#2#,.#,0# 647!78 )91#/!).):
$#,0#; &# ) -%$9!,)3-<)% 4!,#)1 α1u1 + · · ·+ αnun = 0 &5% =%1 &)0!&=#!0)
'(),.% α1 = · · · = αn = 0.
Observação 4.2 >%0# '(# &# α1 = · · ·= αn = 0 #,0<)% α1u1 + · · · +
αnun = 0, 2%15#$8 ) 1#-5?21%-) ,#$ &#$21# 5# /5)4!.)7 @)&0) /#1 '(#8
2%1 #A#$24%8 #$ R
2
0#$%& (0, 0) = 1(1, 1) + 1(−1,−1).
Observação 4.3 B ,%3-<)% .# !,.#2#,.+#,-!) 4!,#)1 2)1) ) &#'*(+#,-!)
u1, . . . , un #'(!/)4# ) .!"#1 '(# &# βi 6= 0 2)1) )4C($ i ∈ {1, . . . , n}
#,0<)% β1u1 + · · · + βnun 6= 0.
Definição 4.4 !"#$%& '(# ($) &#'*(+#,-!) u1, . . . , un .# ($ #&2)3-%
/#0%1!)4 V 5# 4!,#)1$#,0# .#2#,.#,0# 647.78 )91#/!).)$#,0#; &# ,<)% =%1
4!,#)1$#,0# !,.#2#,.#,0#7
Observação 4.5 B .#D,!3-<)% .# .#2#,.+#,-!) 4!,#)1 2)1) ) &#'*(+#,-!)
u1, . . . , un 5# #'(!/)4#,0# ) .!"#1 '(# 5# 2%&&5?/#4 #,-%,01)1 ,5($#1%&
1#)!& α1, . . . , αn ,<)% 0%.%& ,(4%& 0)!& '(# α1u1 + · · · + αnun = 0.
Exemplo 4.6 O,u1, . . . , un ⊂ V 5# ($) &#'*(+#,-!) 47.78 %,.# O 5# %
#4#$#,0% ,#(01% .% #&2)3-% /#0%1!)4 V.
=4*(4 /&',;%4' 6#& 1O + 0u1 + · · · + 0un = O.
Exemplo 4.7 E#1!D'(# &# ) &#'*(+#,-!) (1, 1, 1), (1, 1, 0), (1, 0, 0) 5# 4!,#:
)1$#,0# !,.#2#,.#,0# #$ R
3.
 !"! #$%&'()*+
,
-' . ./.012'3 !
"
# $%&'()* +&%(,'-% ./-() )0-* -) $*))"1+&() )*2/3'0*&) 4&
α(1, 1, 1) + β(1, 1, 0) + γ(1, 0, 0) = (0, 0, 0).
5)6* &./(+-2& - %&)*2+&% * )()6&7-



α + β + γ = 0
α + β = 0
γ = 0,
./& $*))/( '*7* "/8('- )*2/3'0-*9 α = β = γ = 0. :*;*9 - )&.</=&8'(- -'(7-
"& 2>(>>
Exemplo 4.8 !"#$%&'& !# (&)!'&# &* R3 %+%!# ,!'
u1 = (x1, y1, z1), u2 = (x2, y2, z2) & u3 = (x3, y3, z3).
-".!")'& /*+ .!"%$0.1+! "&.&##2+'$+ & #/3.$&")& ,+'+ 4/& !# (&)!'&#
u1, u2, u3 #&5+* 6$"&+'*&")& $"%&,&"%&")&#7
?&@-7*)9 *) +&6*%&) -'(7- )&%0-* 2>(> )& & )*7&86& )& α1u1+α2u2+α3u3 = 0
-$%&)&86-% '*7* "/8('- )*2/3'0-* α1 = α2 = α3 = 0. 5)6* "& &./(+-2&86& - ./&
* )()6&7- 


α1x1 + α2x2 + α3x3 = 0
α1y1 + α2y2 + α3y3 = 0
α1z1 + α2z2 + α3z3 = 0
$*))/- )*2/3'0-* "/8('- &9 '*7* )& )-A&9 ()6* "& &./(+-2&86& ./& - 7-6%(B


x1 x2 x3
y1 y2 y3
z1 z2 z3


$*))/- 4&6&%7(8-86& 4(C&%&86& 4& B&%*> D*6& ./& -) '*2/8-) 4&)6- 7-6%(B
)0-* C*%7-4-) $&2*) '*&,'(&86&) 4& u1, u2 & u3. E 7&)7* %&)/26-4* +-2& )&
'*2*'-%7*) *) '*&,'(&86&) 4*) +&6*%&) u1, u2 & u3 '*7* 2(8F-)> G*% ./=&H
 ! !"
#
$%&'( )* +,",-+
.
,- $! '$-,!/
Exerćıcio 4.9 !"!#$% % &%'(!)*+% "' +%)",*-&( -!.-,(/( -( %0%'1,(
-!*%+$(+ 1-+- "'- )%23"4%!#$- #(' n 5%*(+%) &( Rn.
Exemplo 4.10 6%+$72"% )% -) '-*+$8%)
(
1 0
0 1
)
,
(
1 1
0 1
)
,
(
0 1
0 0
)
)9-( ,$!%-+'%!*% $!&%1%!&%!*%) %' M2.
"#$%&#'($) *) )$+&,%-$') .'
α
(
1 0
0 1
)
+ β
(
1 1
0 1
)
+ γ
(
0 1
0 0
)
=
(
0 0
0 0
)
,
/&' '/&01*+' * (
α + β β + γ
0 α + β
)
=
(
0 0
0 0
)
,
/&' 2$))&0 %$($ )$+&,%-*$ (α, β, γ) = (α, −α,α) 2*#* /&*+/&'# α ∈ R. 3'))*
4$#(*5 * )'/6&7'8%0* .' (*9#0:') .*.* ;' +08'*#('89' .'2'8.'89'5 <*)9*8.$
9$(*#5 2$# '='(2+$5 α = 1, β = −1 ' γ = 1.
Exemplo 4.11 6%+$72"% )% -) :"!;#9(%) %$) % )'8 )9-( ,<&< %' C1(R; R).
>$($ %$) ' )'8 )-*$ 4&8,%-$') .'?80.*) '( R, * %$(<08*,%-*$ 8&+*
α %$)+β )'8 = 0
)0@80?%* /&' α %$) x + β )'8 x = 0 2*#* 9$.$ x ∈ R. A( 2*#90%&+*#5 2*#*
x = 0 1'($) /&' α = 0 ' 2*#* x = π/2, 1'( β = 0. "$#9*89$5 %$) ' )'8
)-*$ +B0BB
Exemplo 4.12 6%+$72"% )% -) :"!;#9(%) %$)2, )'8 2, 1 )9-( ,$!%-+'%!*%
&%1%!&%!*%) %' C1(R; R).
 !"! #$%#$&'()('* !
"#$#
1 − %#&2 x − &'( 2x = 0, )*+* ,#-# x ∈ R,
+'&./,* 0.' *& 1.(2%3#'& *%4$* &3*# /5-55
Exerćıcio 4.13 !"#$ f(x) = %#& 2x, g(x) = %#&2 x ! h(x) = &'( 2x,
x ∈ R. %&'()! *+! f, g, h ',#& -./!#)$!/(! 0!1!/0!/(!' !$ C1(R; R).
4.2 Propriedades
Proposição 4.14 ! u1, . . . , un ',#& -202 !$ +$ !'1#34& 5!(&).#- V
!/(,#& 1!-& $!/&' +$ 0!'(!' 5!(&)!' '! !'4)!5! 4&$& 4&$6./#34,#& -.7
/!#) 0&' &+()&'2
Prova: 6+'%4&*$#& $#&,+*+ 0.' &' u1, . . . , un &3*# /4('*+$'(,' -')'(-'(,'&
'(,3*# '74&,'$ j ∈ {1, . . . , n} ' (8.$'+#& +'*4& α1, . . . , αn−1 ,*4& 0.'
uj = α1u1 + · · · + αj−1uj−1 + αjuj+1 + · · · + αn−1un.
"#$# u1, . . . , un &3*# /5-5 '74&,'$ (8.$'+#& +'*4& β1, . . . , βn (3*# ,#-#&
(./#& ,*4& 0.' β1u1 + · · ·+βnun = 0. 9'&&' $#-#: '74&,' j ∈ {1, . . . , n} ,*/
0.' βj 6= 0 ': *&&4$:
uj = −
β1
βj
u1 − · · · −
βj−1
βj
uj−1 −
βj+1
βj
uj+1 − · · · −
βn
βj
un.
Proposição 4.15 ! u1, . . . , un !$ V ',#& -202 !/(,#& *+#-*+!) '!*8+9!/7
4.# :/.(# 0! 5!(&)!' 0! V *+! &' 4&/(!/;#< (#$6=!$ '!)=# -2022
Prova: ;*$#& $#&,+*+ 0.' &' u1, . . . , un, un+1, . . . , um ∈ V &3*# ,*4& 0.'
u1, . . . , un &3*# /5-5 '(,3*# u1, . . . , un, un+1, . . . , um ,*$<8'$ &3*# /4('*+$'(,'
-')'(-'(,'&5
 ! !"
#
$%&'( )* +,",-+
.
,- $! '$-,!/
"#$# %&'()%$ *+,$%-#( -%.'( β1, . . . , βn */.# )#0#( *,1#( ).'( 2,% β1u1+
· · · + βnun = 0, 3#0%$#( %(4-%5%-
β1u1 + · · · + βnun + 0un+1 + · · · + 0um = 0
(%*0# 2,% *%(). +,1)'$. %&3-%((/.# *%$ )#0#( #( 4#%64'%*)%( (/.# *,1#(7
Proposição 4.16 ! u1, . . . , un, un+1, . . . , um "#$% &'(!$)*!(+! '(,!-!(.
,!(+!" !* /* !"-$01% 2!+%)'$& V !(+#$% 3/$&3/!) "/4"!35/6!(1'$ ,!"+!"
2!+%)!" +$*47!* 7! &'(!$)*!(+! '(,!-!(,!(+!8
Prova: 8.(). $#()-.- 2,% (% u1, . . . , un, un+1, . . . , um (/.# 1'*%.-$%*)% '*9
0%3%*0%*)%( %*)/.# u1, . . . , un ).$:+%$ (/.#7
;,3#*<. 2,% β1u1 + · · · + βnun = 0. =.( 4#$#
β1u1 + · · · + βnun = β1u1 + · · · + βnun + 0un+1 + · · · + 0um = 0
% %()%( 5%)#-%( (/.# 17'7> (%?,% 2,% β1 = · · · = βn = 0.
Proposição 4.17 ! u1, . . . , un "#$% &8'8 !* /* !"-$01% 2!+%)'$& V !
u1, . . . , un, un+1 "#$% &8,8 !(+#$% un+1 7! 1%*4'($01#$% &'(!$) ,! u1, . . . , un.
Prova: @&'()%$ β1, . . . , βn+1 */.# )#0#( *,1#( ).'( 2,%
β1u1 · · · + βnun + βn+1un+1 = 0.
A?#-.> (% βn+1 = 0 %*)/.# . %&3-%((/.# .4'$. 64.-'.
β1u1 · · · + βnun = 0.
B-.> #( 5%)#-%( u1, . . . , un (/.# 17'7 %> .(('$> 0%5%-+C.$#( )%- ).$:+%$ β1 =
· · · = βn = 0. D$. 4#*)-.0'E4/.#7
Proposição 4.18 !9$* u1, . . . , un 2!+%)!" &8'8 !* /* !"-$01% 2!+%)'$&
V. :(+#$% 1$,$ 2!+%) v ∈ [u1, . . . , un] "! !"1)!2! ,! *$(!')$ 7/('1$ 1%*%
v = α1u1 + · · · + αnun.
 !"! #$#%&
'
(&()* !
Prova:
"#$%# &'$%(#( )*+ $+ α1u1 + · · · + αnun = β1u1 + · · · + βnun +,%-#'
αj = βj, j = 1, . . . , n.
.+&'$
(α1 − β1)u1 + · · · + (αn − βn)un = 0
+ /'&' u1, . . . , un $-#' 0121 +,%-#' αj − βj = 0, 2$%' 3+ αj = βj, 4#(# %'5'
j = 1, . . . , n.
4.3 Exerćıcios
Ex. 4.19 !"#$%&!' !( )*+* &( +,- #.!/- *0*#1,' -! , -&0),/2&/., S
+, !-3*4), 5!.,"#*6 V 7! 68#8 ,& 68+8
61 S = {(1, 2), (−3, 1)} , V = R2.
71 S =
{
1 + t − t2, 2 + 5t − 9t2
}
, V = P2(R).
!1 S =
{(
−1 1
0 0
)
,
(
2 0
−1 0
)}
, V = M2.
 1 S = {(1, 2, 2,−3), (−1, 4,−2, 0)} , V = R4.
81 S =





1 2 0
3 0 1
0 0 2

 ,


−1 −1 −1
0 0 0
1 1 1

 ,


0 0 0
10 5 7
−1 0 1





, V = M3.
91 S = {1, $+, x, /'$ x} , V = C∞(R, R).
:1 S =
{
1, $+, 2x, /'$2 x
}
, V = C∞(R, R).
;1 S = {ex, e−x} , V = C∞(R, R).
<1 S = {xex, x} , V = C∞(R, R).
 !"
#
$%&'( )* +,",-+
.
,- $! '$-,!/
Ex. 4.20 !"# S = {u, v, w} $% &'("$()' *+,+ !% V. -!.,/0$! 1! '1
&'("$()'1 #2#,3' 14#' *+,+ '$ *+5++
6+ S1 = {u, u + v, u + v + w};
7+ S2 = {u − v, v − w,w − u};
8+ S3 = {u + v, u + v + w,w}.
Ex. 4.21 !"#% f, g ∈ C1((a, b); R). 9'1).! 0$! 1! !3,1),. x ∈ (a, b) )#*
0$! f(x)g ′(x) 6= f ′(x)g(x) !()4#' f ! g 14#' *+,++
Caṕıtulo 5
Base, Dimensão e Coordenadas
5.1 Base
 
 !"#$%! &' (%)' &' *+ '),%"#! -'.!/0%1 2' +*0.! )0+,1')3 41% #! )0).'
'+ ')#!15'/ *+ #! 6* .! &' 7'/%&!/') 8*' )'6% ! +' !/ ,!))29-'1: 0).!
2': *+ #! 6* .! 8*' 7'/' ! '),%"#!: +%) 8*' )' &').' #! 6* .! ;!/ )*(./%29&!
8*%18*'/ '1'+' .!: ! 8*' /').% $%! 7'/% +%0) ! '),%"#! .!&!3
<'6%+!) % &'= 0"#$%! ,/'#0)% &' (%)'3
Definição 5.1 !"# V 6= {0} $% !&'#()* +!,*-.#/ 01.,#%!1,! 2!-#3*4
5%# 6#&! 3! V 7! $%# &!89$:!1).# 3! +!,*-!& /.1!#-%!1,! .13!'!13!1,!&
B 3! V 8$! ,#%67!% 2!-# V.
Exemplo 5.2 ;& +!,*-!& 3! B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} <*-%#% $=
%# 6#&! 3! R
3.
<>'?)' ;%#01+' .' 8*' !) -'.!/') &' B )$%! 1303 ' 8*' .!&! (x, y, z) ∈ R3 )'
')#/'-' #!+! (x, y, z) = x(1, 0, 0) + y(0, 1, 0) + z(0, 0, 1).
Exemplo 5.3 ;& +!,*-!& e1, . . . , en ∈ Rn *13! e1 = (1, 0, . . . , 0), e2 =
(0, 1, 0, . . . , 0), . . . , en = (0, . . . , 0, 1) <*-%#% $%# 6#&! 3! R
n.
@A
 !!"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
Ex. Resolvido 5.4 !"#$% &'% (1, 1) % (1,−1) (!$)*) ')* +*"% ,%
R
2.
Resolução: "# $%&'()* +*),%-% ./& &),&) 0&,*%&) )1-* 23(3 & ./& ,*4* $*5,*
4& R
2
)& &)'%&0& '*+* '*+6(5-7'1-* 2(5&-% 4& (1, 1) & (1,−1). 8* &5,-5,*9
)& +*),%-%+*) ./& ,*4* $*5,* 4& R
2
)& &)'%&0& ,% )*-%.$* /'-.0* '*+*
'*+6(5-7'1-* 2(5&-% 4& (1, 1) & (1,−1) :"- &),-%&+*) +*),%-54* -) 4/-) $%*;
$%(&4-4&) -* +&)+* ,&+$*3 <=*% ./>&?@
A&:- (x, y) ∈ R2. B 5*))* $%*62&+- )& %&)/+& &+ +*),%-% ./& &C(),& /+
"/5('* α ∈ R & /+ "/5('* β ∈ R )-,()D-E&54* (x, y) = α(1, 1) + β(1,−1) =
(α + β,α − β). #),- "/2,(+- &C$%&))1-* "& &./(0-2&5,& -* )&F/(5,& )(),&+-
2(5&-% {
α + β = x
α − β = y.
G&)*20&54* * )(),&+- *6,&+*) /+- "/5('- )*2/7'1-* 4-4- $*% α = (x + y)/2
& β = (x − y)/2. ¤
Exemplo 5.5 1" )*#$.2%" %)
B =
{(
1 0
0 0
)
,
(
0 1
0 0
)
,
(
0 0
1 0
)
,
(
0 0
0 1
)}
(!$)*) ')* +*"% ,% M2.
Exerćıcio 5.6 3%$.4&'% "% !" %5%)%-#!" ,% B = {1 + x, 1 − x, 1 − x2}
(!$)*) ')* +*"% ,% P2(R).
Proposição 5.7 6%7* {u1, . . . , un} ')* +*"% ,% V. 8-#9*! {u1, . . . , un−1}
-9*! /% ')* +*"% ,% V.
Prova: A& {u1, . . . , un−1} D*))& /+- 6-)& 4& V &5,1-* &C(),(%(-+ αj ∈ R,
j = 1, . . . , n − 1 ,-() ./&
un = α1u1 + · · · + αn−1un−1,
 !"! #$%&'(
)
*+ !
"#$% &'(
α1u1 + · · · + αn−1un−1 − un = 0,
)%*$+,-".'*-% % /,$% -' 01' u1, . . . , un #2,% 3"*',+4'*$' "*-'5'*-'*$'#6
Teorema 5.8 !"! #$%&'(! )#*!+,&- V 6= {0} ./,*&0#/*# 1#+&"! &"0,*#
20& 3&$#4 50 !2*+&$ %&-&)+&$6 78& 20& $#9:2;#/(,& "# )#*!+#$ -4,4 "#
V <!+0&"& %!+ 1#+&"!+#$4
Prova: 7%4% V 6= {0} &' 8*"$,4'*$' 9'+,-% ':"#$'4 u1, . . . , un ∈ V $,"#
01' V = [u1, . . . , un]. ;' u1, . . . , un /%+'4 36"6( '*$2,% '#$, #'0<1='*)", &' 14,
>,#' -' V ' *2,% ?&, *,-, 4,"# , #'+ 5+%@,-%6
;15%*?,4%# 01' u1, . . . , un #'A,4 36-66 7%4% V 6= {0}, ':"#$' j ∈
{1, . . . , n} $,3 01' uj 6= 0. B%+ #"453")"-,-'( 5%-'4%# #15%+ 01' u1 6= 0.
C9%+,( #' $%-% uj, j = 2, . . . , n 51-'+ #' '#)+'@'+ )%4% )%4>"*,D)2,% 3"*',+
-' u1 '*$2,% V = [u1] ' u1 &' 14, >,#' -' V. 7,#% "#$% *2,% %)%++,( &' 5%+01'
':"#$' ,3914 uj, )%4 2 ≤ j ≤ n $,3 01' u1, uj #2,% 36"66 B%+ #"453")"-,-'(
#15%*?,4%# 01' #'A, % u2, "#$% &'( u1, u2 #2,% 36"66 E'4( #' $%-%# %# @'$%+'#
u3, . . . , un /%+'4 )%4>"*,D)2%'# 3"*',+'# -' u1 ' u2 '*$2,% V = [u1, u2] '
u1, u2 /%+4,4 14, >,#' -' V. B%-'4%# +'5'$"+ '#$' 5+%)'##% ' )%4% %
*&14'+% -' '3'4'*$%# -' L = {u1, . . . , un} &' 8*"$%( '3' 8*-,6 F'##' 4%-%(
':"#$' 14, #'0<1='*)", -' @'$%+'# 36"6 -'*$+' %# @'$%+'# L 01' 9'+, V. G#$,
#'0<1='*)", /%+4, 14, >,#' -' V.
5.2 Dimensão
Teorema 5.9 50 20 #$%&'(! )#*!+,&- V 6= {0} ./,*&0#/*# 1#+&"! *!"&
3&$# %!$$2, ! 0#$0! /820#+! "# #-#0#/*!$4
Prova: ;'A,4 u1, . . . , un ' v1, . . . , vm >,#'# -' 14 '#5,D)% @'$%+",3 8*"$,H
4'*$' 9'+,-% V. ;15%*?,4%# 01' n > m ' 4%#$+'4%# 01' "#$% "453"),+&,
01' u1, . . . , un #2,% 36-6( % 01' )%*$+,+", % /,$% -' /%+4,+'4 14, >,#'6
 ! !"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
"#$# #% &'(#)'% v1, . . . , vm *')+$ V ,#-'$#% '%.)'&') ,+)+ .+-+ 1 ≤
j ≤ n,
uj = α1jv1 + · · · + αmjvm.
/%%0$1 + .#$203+4.5+# 603'+) 376+ x1u1 + · · · + xnun = 0 8' '970&+6'3(' +
x1
(
m∑
i=1
αi1vi
)
+ · · · + xn
(
m∑
i=1
αinvi
)
= 0,
#7 +03-+1 (
n∑
j=1
xjα1j
)
v1 + · · · +
(
n∑
j=1
xjαmj
)
vm = 0.
"#$# v1, . . . , vm %5+# 6:0: '3(5+#
∑n
j=1 xjαij = 0 ,+)+ (#-# 1 ≤ i ≤ m. ;%(+%
m '97+4.5#'% )',)'%'3(+$ 7$ %0%('$+ 603'+) <#$#*='3'# .#$ n 03.8#*30(+%:
"#$# n > m, '>0%(' 7$+ %#674.5+# 35+# ()0&0+61 0%(# 8'1 7$+ %#674.5+# x1, . . . , xn
#3-' ,'6# $'3#% 7$ xj 8' -0?')'3(' -' @')#: /%%0$1 u1, . . . , un %5+# 6:-:1 7$+
.#3()+-04.5+#:
Definição 5.10 !"# V $% !&'#()* +!,*-.#/ 01.,#%!1,! 2!-#3*4 !
V = {0} 3!01.%*& # 3.%!1&5#* 3! V )*%* &!13* 0. ! V 6= {0} 3!01.%*&
# 3.%!1&5#* 3! V )*%* &!13* * 16$%!-* 3! !/!%!1,*& 3! $%# 7#&!
8$#/8$!- 3! V. 9&#-!%*& * &6:%7*/* -0$V '#-# 3!&.21#- # 3.%!1&5#*
3! V.
Definição 5.11 ! $% !&'#()* +!,*-.#/ 15#* 6! 01.,#%!1,! 2!-#3* 3.;!<
%*& 8$! V '*&&$. 3.%!1&5#* .101.,#4
Proposição 5.12 =*3* !&'#()* +!,*-.#/ 3! 3.%!1&5#* .101.,# '*&&$.
$%# .101.3#3! 3! +!,*-!& /.1!#-%!1,! .13!'!13!1,!&4
Prova: A'B+ V 7$ '%,+4.# &'(#)0+6 -' -0$'3%5+# 03C30(+: "6+)+$'3(' V 6=
{0}. A'6'.0#3' u1 ∈ V, u1 6= 0. "#$# V 35+# 8' C30(+$'3(' *')+-#1 V 6= [u1].
 !"! #$%&'(
)
*+ !
"##$%& '()*%(# +(%,- u2 ∈ V +,. /0* u2 6∈ [u1]. 1*#+, 2(-%,& (# 3*+(-*#
u1 * u2 #4,( .$5*,-%*5+* $5)*'*5)*5+*#6
70'(58, /0* +*58,%(# *59(5+-,)( 3*+(-*# u1, . . . , un ∈ V .$5*,-%*5+*
$5)*'*5)*5+*#6 :(%( V 54,( ;* <5$+,%*5+* =*-,)(& V 6= [u1, . . . , un] *&
,##$%& ;* '(##;>3*. *#9(.8*- un+1 ∈ V +,. /0* un+1 6∈ [u1, . . . , un], $#+( ;*& (#
3*+(-*# u1, . . . , un, un+1 ∈ V #4,( .$5*,-%*5+* $5)*'*5)*5+*#6
?% -*#0%(& *@$#+* *% V 0%, #*/A0B*59$, $5<5$+, )* 3*+(-*# .$5*,-%*5+*
$5)*'*5)*5+*#6
" #*=0$5+* '-('(#$C94,( ;* 0% -*#0.+,)( ), '-(3, )( +*(-*%, D6!6
Proposição 5.13 ! "! #$%&'() *#+),-&. /# /-!#0$1&) m 2"&.2"#,
$#23"4#0(-& /# *#+),#$ ()! !&-$ /# m #.#!#0+)$ 5# .-0#&,!#0+# /#6
%#0/#0+#7
Corolário 5.14 8)/) $"9#$%&'() *#+),-&. /# "! #$%&'() *#+),-&. /# /-6
!#0$1&) :0-+& +&!95#! +#! /-!#0$1&) :0-+&7
Prova: 7*E, V 0% *#',C9( 3*+(-$,. )* )$%*5#4,( <5$+, * W 0% #0F*#',C9(
3*+(-$,. )* V. 7* W +$3*##* )$%*5#4,( $5<5$+,& '*., '-('(#$C94,( D6GH& *@$#I
+$-$, 0%, $5<5$),)* )* 3*+(-*# .$5*,-%*5+* $5)*'*5)*5+*# *% W. :(%(
*#+*# 3*+(-*# +,%F;*% #4,( .$5*,-%*5+* $5)*'*5)*5+*# *% V, ( 5;0%*-( )*.*#
)*3*-$, #*- %*5(- )( /0* , )$%*5#4,( )* V J'*., '-('(#$C94,( D6GKL6 M%,
9(5+-,)$C94,(6
Corolário 5.15 ;# V 5# "! #$%&'() *#+),-&. n6/-!#0$-)0&. # u1, . . . , un
$1&) *#+),#$ /# V .-0#&,!#0+# -0/#%#0/#0+#$ #0+1&) #$+#$ *#+),#$ <),6
!&! "!& 9&$# /# V.
Exemplo 5.16 )$%Rn = n.
Exemplo 5.17 = /-!#0$1&) /# P(R) 5# -0:0-+&7 >#?& ) #@#!%.) A7BC7
Exemplo 5.18 )$%Pn(R) = n + 1.
 ! !"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
"#$%# &'%#( )*+ '$ ,'-.&/'0.'$ 1, x, . . . , xn 1'(0#0 *0# 2#$+ 3+ Pn(R).
Exemplo 5.19 3.0Mm×n = mn.
4'%+ )*+ #$ 0#%(.5+$
Ak,l = (δ
k,l
i,j )1≤i≤m
1≤j≤n
,
k = 1, . . . , m, l = 1, . . . , n '&3+
δk,li,j =
{
1 $+ (i, j) = (k, l)
0 $+ (i, j) 6= (k, l)
1'(0#0 *0# 2#$+ 3+ Mm×n.
Exerćıcio 5.20 !"#$%&'() !) $&*(+,) !(& #(-."/$& 01(!.(!(& $ &"2
#3$-.",(& !$ ).!$# n 3$ n(n + 1)/2.
Teorema 5.21 (Completamento) 4$5( V 1# $&*(+,) 6$-)."(7 !$ !"2
#$%&'() n. 4$ )& 6$-).$& u1, . . . , ur &'() 78"8 $# V ,)# r < n $%-'()
$9"&-$# ur+1, . . . , un -("& 01$ u1, . . . , ur, ur+1, . . . , un :).#(# 1#( ;(&$
!$ V.
Prova: 6'0' r < n +7.$%+ ur+1 ∈ V %#- )*+ u1, . . . , ur, ur+1 $8#' -9.9: ,'.$
;#$' ;'&%(<#(.' '$ =+%'(+$ u1, . . . , ur 1'(0#(.#0 *0# 2#$+ 3+ V, ' )*+ <+
.0,'$$<>=+- ,'.$ 3.0V = n > r.
?+ r + 1 = n +&%8#' u1, . . . , ur, ur+1 1'(0#0 *0# 2#$+ 3+ V.
?+ r+1 < n +&%8#' <+ ,'$$<>=+- +&;'&%(#( ur+2 ∈ V %#- )*+ u1, . . . , ur, ur+1,
ur+2 $8#' -9.9: ,'.$ ;#$' ;'&%(<#(.' # $+)@*/+&;.# u1, . . . , ur, ur+1 $+(.# *0# 2#$+
3+ V, ' )*+ <+ .0,'$$<>=+- ,'.$ 3.0V = n > r + 1.
A+,+%.&3' '$ #(B*0+&%'$ #;.0#: +&;'&%(#0'$ =+%'(+$ ur+1, ur+2, . . . ,
ur+k, '&3+ r + k = n, 3+ 1'(0# )*+
u1, . . . , ur, ur+1, . . . , ur+k
$8#' -9.9 +: ;'0' 3.0V = n = r + k, $+B*+ )*+ +$%# $+)@*/+&;.# 3+ =+%'(+$ <+
*0# 2#$+ 3+ V )*+ ;'&%<+0 '$ =+%'(+$ u1, . . . , ur.
 !"! #$%&'(
)
*+ #& (+%* #& (,-&(.*/0+( 1&2+3$*$( !
Exemplo 5.22 !"#!$%& '() *)+& ,# R3 "#!$&!,# # -&$#% (1, 1,−1).
"#$# % &'$()*+%# &( R
3
,( -./(*0 1.(2'*%$#* ()2#)-.%. &#'* 3(-#.(*0 (a, b, c),
(x, y, z), 45( 65)-%$()-( 2#$ (1, 1,−1) *(6%$ 78'88 9#.,($0 1(7# (:($17#
;8<0 *%=($#* 45( '*-# ,( (45'3%7()-( %# &(-(.$')%)-( &(


1 a x
1 b y
−1 c z


45( ,( &%&# 1#. x(b + c) − y(a + c) + z(b − a) *(6% &'>(.()-( &( ?(.#8 @,%
5$% ')A)'&%&( &( 1#**'='7'&%&(* 1%.% 45( '*-# %2#)-(B2%8 9#. (:($17#0
-#$%)&# (a, b, c) = (0, 1, 1) ( (x, y, z) = (0, 0, 1).
5.3 Dimensão de Soma de Subespaços Veto-
riais
Proposição 5.23 !"# V $% !&'#()* +!,*-.#/ 0! 0.%!1&2#* 31.,#4 !
U ! W &2#* &$5!&'#()*& +!,*-.#.& 0! V !1,2#*
 !"U ∩ W + !" (U + W) = !"U + !"W #$%&'(
Prova: )*"+,*-.* /0 0 1.+*123450 * ." *123450 6*/0,!37 * !"*81930
:8!/3 /*" /3"+;*" !"*81930 :8!/3%
<*=3" v1, . . . , vm *7*"*8/01 * ."3 +31* * U∩W. >0"0 *1/*1 6*/0,*1
1930 7%!% * 2*,/*85*" 3 U, 2*70 /*0,*"3 $%&?@ *A!1/*" u1, . . . , up ∈ U
/3!1 -.* u1, . . . , up, v1, . . . , vm B0,"3" ."3 +31* * U. C0, 0./,0 73 0@
01 6*/0,*1 v1, . . . , vm /3"+;*" 2*,/*85*" 3 W * 2*70 "*1"0 /*0,*"3 ;*
2011;D6*7 *8508/,3, w1, . . . , wq ∈ W * "0 0 -.* w1, . . . , wq, v1, . . . , vm
B0,"*" ."3 +31* * W.
>0" 3 80/345930 .13 3@ /*"01 !"U ∩ W = m, !"U = m + p *
 !"W = m + q. <*8 0 311!"@ 3 :" * "01/,3,"01 -.* $%&' ;* 6;37! 3@ ;*
 ! !"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
"#$#%%&'()* #+ "' ,#(-'-#+ %./$)#"0# 1*%0('( 2.# -)1 (U+W) = m+p+q.
3'(' 0'"0*+ 4'%0' 1*%0('(1*% 2.# *% ,#0*(#%
u1, . . . , up, w1, . . . , wq, v1, . . . , vm 5 6! 7
8*(1'1 .1' 4'%# -# U + W.
9*%0(#1*% :()1#)('1#"0# 2.# #;#% <#('1 U + W : -'-* v ∈ U + W
#=)%0#1 u ∈ U # w ∈ W 0')% 2.# v = u + w. >*1* u &# .1' $*1?
4)"'@$A'* ;)"#'( -# u1, . . . , up, v1, . . . , vm # w &# .1' $*14)"'@$A'* ;)"#'( -#
w1, . . . , wq, v1, . . . , vm %#<.# 2.# v = u + w &# .1' $*14)"'@$A'* ;)"#'( -#
u1, . . . , up, v1, . . . , vm,1 , . . . , wq. 3*(0'"0*+
U + W = [u1, . . . , up, v1, . . . , vm,1 , . . . , wq].
B#()/2.#1*% 2.# *% ,#0*(#% #1 6! %A'* ;6)66 C.:*"D' 2.#
α1u1 + · · · + αpup + β1w1 + · · · + βqwq + δ1v1 + · · · + δmvm = 0, 5 6!E7
*. %#F'
U ∋ α1u1 + · · · + αpup + δ1v1 + · · · + δmvm = −β1w1 − · · · − βqwq ∈ W.
G*<*+
−β1w1 − · · · − βqwq ∈ U ∩ W = [v1, . . . , vm].
>*"%#2H.#"0#1#"0#+ #=)%0#1 γ1, . . . , γm 0')% 2.#
−β1w1 − · · · − βqwq = γ1v1 + · · · + γmvm,
*. %#F'+
β1w1 + · · · + βqwq + γ1v1 + · · · + γmvm = 0.
>*1* w1, . . . , wq, v1, . . . , vm %A'* ;6)6+ :*)% 8*(1'1 .1' 4'%# -# W, %#<.#?%#
2.# γ1 = · · · = γm = β1 = · · · = βq = 0. I%%)1+ ' #2.'@$A'* 6!E %# (#-.J '
α1u1 + · · · + αpup + δ1v1 + · · · + δmvm = 0
 !"! #$%&'(
)
*+ #& (+%* #& (,-&(.*/0+( 1&2+3$*$( !
" #$%$ u1, . . . , up, v1, . . . , vm &'($ )*+*, -$+& .$/%(% 0%( 1(&" 2" U, &"30"4&"
50"
α1 = · · · = αp = δ1 = · · · = δm = 0,
$0 &"6(, $& 7"8$/"& 2" *9 &'($ )+:"(/%":8" +:2"-":2":8"&*
Corolário 5.27 !"# U $% &$'!&(#)*+ ,!-+./#0 1! $% !&(#)*+ ,!-+./#0
1! 1/%!2&3#+ 42/-# V. ! 2+%U = 2+%V !2-3#+ U = V.
Prova: ;0-$:<( 50" "=+&8( u1 ∈ V #$% u1 6∈ U. >$)$50" W = [u1]. >$%$
U ∩ W = {0} " 2+%W = 1, &"30" 2( -/$-$&+?#'($ *9! 50"
2+% (U + W) = 2+%U + 1 = 2+%V + 1 > 2+%V.
@% (1&0/2$ -$+& 2+% (U + W) ≤ 2+%V.
Observação 5.28 5+-! 6$! &! V, U ! W &3#+ *+%+ 2# (.+(+&/)*3#+ 789:
! &! #0;!% 1+ %#/& -/,!.%+& V = U + W ! 2+%U + 2+%W > 2+%V
!2-3#+ U ∩ W 6= {0}, /&-+ ;!< # &+%# U + W 23#+ ;! 1/.!-#8
A"%, &" .$&&" U ∩ W = {0} ":8'($ -")( -/$-$&+?#'($ *9! 8"/BC(%$&
0 = 2+%U ∩ W = 2+%U + 2+%W − 2+% (U + W)
= 2+%U + 2+%W − 2+%V > 0,
0% (1&0/2$*
Exemplo 5.29 !"#% U = {p(x) ∈ P3(R); p(0) = p(1) = 0} ! V =
{p(x) ∈ P3(R); p(−1) = 0}. =2*+2-.! $%# '#&! 1! U, V, U ∩ V ! U + V.
U : D"%$&
p(x) = a0 + a1x + a2x
2 + a3x
3 ∈ U ⇐⇒ p(0) = p(1) = 0
 ! !"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
⇐⇒
{
a0 = 0
a0 + a1 + a2 + a3 = 0
⇐⇒ p(x) = −(a2 + a3)x + a2x2 + a3x3 = a2(x2 − x) + a3(x3 − x).
"#$$# %&'&( U = [x2−x, x3−x] # #$)#$ *&+,-.&%,&$ $/0& +1,1 *&,$ 2&%&
20'0 3% )#% 3% 4503 ',$),-)& '& &3)5&( -#-63% *&'# $#5 %73+),*+&
'& &3)5&1 8$$,%( x2 − x # x3 − x 9&5%0% 3%0 :0$# '# U.
V :
p(x) = a0 + a1x + a2x
2 + a3x
3 ∈ V
⇐⇒ p(−1) = 0 ⇐⇒ a0 − a1 + a2 − a3 = 0
⇐⇒ p(x) = a0 + (a0 + a2 − a3)x + a2x2 + a3x3
= a0(1 + x) + a2(x
2 + x) + a3(x
3 − x).
"#$$# %&'&( V = [1 + x, x2 + x, x3 − x] # #$)#$ *&+,-.&%,&$ $/0& +1,1
*&,$ 2&%& 20'0 3% )#% 3% 4503 ',$),-)& '& &3)5&( -#-63% *&'#
$#5 3%0 2&%:,-0;2/0& +,-#05 '&$ &3)5&$ '&,$1 <&5)0-)&( 1 + x, x2 + x #
x3 − x 9&5%0% 3%0 :0$# '# V.
U ∩ V :
p(x) = a0+a1x+a2x
2+a3x
3 ∈ U∩V ⇐⇒



a0 = 0
a0 + a1 + a2 + a3 = 0
a0 − a1 + a2 − a3 = 0
⇐⇒
{
a0 = a2 = 0
a1 = −a3
⇐⇒ p(x) = −a1(x3 − x).
=&4&( x3 − x 7# 3%0 :0$# '# U ∩ V.
U + V : >#%&$ ',% (U + V) = 2 + 3 − 1 = 4 = ',%P3(R). <#+0 *5&*&$,;2/0&
 1?@ )#%&$ A3# U + V = P3(R) # *&'#%&$ )&%05 2&%& :0$# &$
*&+,-.&%,&$ 1, x, x2 # x3.
 !"! #$%&'(
)
*+ #& (+%* #& (,-&(.*/0+( 1&2+3$*$( 
Exemplo 5.30 !"#$%!& '! $($%)"! *+,-+ .'/$%!& 01$
U = [(1, 1, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1)]
V = [(1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1)]
U ∩ V = [(1, 0,−1, 0), (0, 1, 0, 1)]
U + V = [(0, 1, 1, 0), (0, 1, 0, 1), (1, 0, 0, 1), (0, 0, 1, 1)]
!"#$%&'"()* &'" )* +"#,-)#"* ,.$(, */,) 0, 1"#-,-" 2,*"* 3,#, )* #"*3".4
5$1)* *'2"*3,6.)* 1"5)#$,$*7 8,#, 5,05) 2,*5, 1"#$%.,# &'" .,-, *"&9':"0.$,
-" 1"5)#"* ,.$(, ;" <7$77
=0,<$*"()* 3#$("$#,("05" 3,#, U> *"
α(1, 1, 0, 0) + β(0, 1, 1, 0) + γ(0, 1, 0, 1) = (0, 0, 0, 0)
"05/,)
(α, α + β + γ,β, γ) = (0, 0, 0, 0)
&'" $(3<$., "( α = β = γ = 0.
!"?,()* ,+)#, ) .,*) -) *'2"*3,6.) V > *"
α(1, 0, 0, 1) + β(0, 1, 0, 1) + γ(0, 0, 1, 1) = (0, 0, 0, 0)
"05/,)
(α, β, γ, α + β + γ) = (0, 0, 0, 0)
&'" $(3<$., "( α = β = γ = 0.
8,**"()* ,+)#, , U ∩ V : *"
α(1, 0, −1, 0) + β(0, 1, 0, 1) = (α,β, −α,β) = (0, 0, 0, 0)
&'" $(3<$., "( α = β = 0.
8"<, 3#)3)*$6./,) 7@A 5"()* -$( (U + V) = 3 + 3 − 2 = 4. B)()
(0, 1, 1, 0), (0, 1, 0, 1), (1, 0, 0, 1), (0, 0, 1, 1) +"#,( U + V *"+'"4*" -) C,5)
-, -$("0*/,) -"*5" *'2"*3,6.) *"# &',5#) &'" C)#(,( '(, 2,*" -" U + V.
B)() , -$("0*/,) -" R
4
5,(2;"( " U + V ⊂ R4, 5"()* 3"<, 3#)3)*$6./,)
 7@D &'" U + V = R4. E)5" &'" "*5, *)(, 0/,) ;" -$#"5,7
 ! !"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
5.4 Coordenadas
"#$%& V '& #()%*+, -#.,/0%1 230.%&#3.# 4#/%5, # B '&% 6%(# 5# V 7,/&%5%
)#1,( -#.,/#( u1, . . . , un. 8,&, B 9# '&% 6%(# 5# V, .,5, #1#&#3., 5# u ∈ V
(# #(+/#-# +,&, α1u1 + · · · + αnun, +,& ,( +,#2+0#3.#( α1, . . . , αn ∈ R.
:#1% )/,),(0*+;%, <=>?@ ,( +,#2+0#3.#( α1, . . . , αn (;%, '30+%&#3.# 5#.#/&0A
3%5,( )#1, -#.,/ u. B(.#( +,#2+0#3.#( (;%, 5#3,&03%5,( +,,/5#3%( 5# u
+,& /#1%*+;%, C% 6%(# B. D#)/#(#3.%/#&,( %( +,,/5#3%5%( 5# u +,& /#1%*+;%,
C% 6%(# +,&,
uB =


α1
=
=
=
αn

 .
Exemplo 5.31 !"#$% &'% !" (%#!$%" (1, 1, 1), (0, 1, 1) % (0, 0, 1) )!$*
+,+ '+, -,"% .% R
3. /01!0#$% ," 1!!$.%0,.," .% (1, 2, 0) ∈ R3 1!+
$%2,314,! 5, -,"% B )!$+,., 6%2!" (%#!$%" ,17+,8
E9% (%6#&,( F'# 50&R
3 = 3. :%/% -#/02+%/ (# ,( -#.,/#( %+0&% 7,/&%&
'&% 6%(# 5# V, 6%(.% -#/02+%/ (# #1#( (;%, 1=0== G.010H%35, , #I#&)1, <=?
-#&,( F'# #(.#( -#.,/#( (;%, 5# 7%., 1=0= ),0( % &%./0H


1 0 0
1 1 0
1 1 1


),(('0 5#.#/&03%3.# 04'%1 % 1 6= 0.
J4,/%@
(1, 2, 0) = α(1, 1, 1) + β(0, 1, 1) + γ(0, 0, 1) = (α,α + β,α + β + γ)
F'# 9# #F'0-%1#3.# %, (0(.#&%



α = 1
α + β = 2
α + β + γ = 0
 !"! #$$%&'()&)* !
"#$% &'#()"%* +,-#."/%, '0 α = 1, β = 1 0 γ = −2. 10++0 2,3,4 %+ ",,530(%3%+
30 (1, 2, 0) ",2 50-%."/%, 6% 7%+0 B +/%, 3%3%+ 8,5


1
1
−2

 .
Exemplo 5.32 !"#$% &'% !" (!)*+,!-*!" 1, x, x2−x .!$-/- '-/ 0/"%1
B, 2% P2(R). 3+4!+#$% /" 4!!$2%+/2/" 2% 1 + x + x
2
4!- $%)/546/! 7/
0/"% B. 3+4!+#$% #/-08%- /" 4!!$2%+/2/" 2%"#% -%"-! (!)*+,!-*! 4!-
$%)/546/! 7/ 0/"% C .!$-/2/ (%)!" (!)*+,!-*!" 1, x % x2.
9%5% :05);"%5 <#0 1, x, x2−x =,52%2 #2% 7%+0 30 P2(R) 7%+>% 2,+>5%5
"%3% p(x) = a0 + a1x + a2x
2 ∈ P2(R) +0 0+"50:0 30 2%(0)5% '#()"% ",2,
",27)(%."/%, -)(0%5 30 1, x 0 x2 − x. ?+>, '0 0<#):%-0(>0 % 2,+>5%5 <#0 %
0<#%."/%, p(x) = α1+βx+γ(x2−x) 8,++#) #2% '#()"% +,-#."/%, (α,β, γ) ∈ R3.
@ 0<#%."/%, %")2% +0 0+"50:0 ",2,
a0 + a1x + a2x
2 = α + (β − γ)x + γx2,
<#0 '0 0<#):%-0(>0 %, +)+>02%



α = a0
β − γ = a1
γ = a2,
<#0 8,++#) #2% '#()"% +,-#."/%, 3%3% 8,5 α = a0, β = a1 + a2, 0 γ = a2.
A,2 )++, 02 2/%,+4 :02,+ <#0 %+ ",,530(%3%+ 30 1+x+x2 ",2 50-%."/%,
6% 7%+0 B +/%, 3%3%+ 8,5 

1
2
1

 .
 ! !"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
"#$% &'% (#) *%+,-(.,# /, 0,1% C 2#*),3, 4#* 1, x % x2 ,1 (##*3%5,3,1 3%
1 + x + x2 1.,# 3,3,1 4#* 

1
1
1

 .
5.5 Exerćıcios
Ex. 5.33 !"#$%&" !' %&(& )' (*+ %&+*+ +! * +),%*-.)-/* B (*
!+0&1%* 2!/*"#&3 V 4! )'& ,&+! (! V.
67 B =
{
1, 1 + t, 1 − t2,1 − t − t2 − t3
}
, V = P3(R).
87 B =
{(
1 1
0 0
)
,
(
2 1
0 0
)
,
(
0 1
1 0
)
,
(
0 0
0 2
)}
, V = M2.
97 B = {(1, 1, 1, 1), (1, 1, 1, 0), (1, 1, 0, 0), (1, 0, 0, 0)} , V = R4.
Ex. 5.34 5-%*-/"&" !' %&(& )' (*+ #/!-+ &,&#6* )'& ,&+! ! & (#7
'!-+8&* (* +),!+0&1%* W (* !+0&1%* 2!/*"#&3 V.
67 W =
{
(x, y, z, t) ∈ R4; x − y = 0 % x + 2y + t = 0
}
, V = R4.
87 W = {X ∈ M2;AX = X} , #53% A =
(
1 2
0 1
)
, V = M2.
97 W = {p ∈ P2(R);p′′(t) = 0,∀t ∈ R} , V = P2(R).
:7 W = {X ∈ M2;AX = XA} , #53% A =
(
1 0
1 1
)
, V = M2.
Ex. 5.35 9&(*+ U, W +),!+0&1%*+ (* !+0&1%* 2!/*"#&3 V (!/!"'#-&":
#; )'& ,&+! ! & (#'!-+8&* (! U.
 ! ! "#"$%
&
'%'() !
 ! "#$ %$&' ' $ ( #')&*$+ (' W.
 ! "#$ %$&' ' $ ( #')&*$+ (' U + W.
 ,! "#$ %$&' ' $ ( #')&*$+ (' U ∩ W. )+& &'-" ).'& /$&+&0
12 U =
{
(x, y, z) ∈ R3; x + y + z = 0
}
, W = {(x, y, 0); x, y ∈ R} , V =
R
3.
32 U = {A ∈ M2; "# (A) = 0} , W = {A ∈ M2;At = −A} , V = M2. "# (A)
4' $ &+#$ (+& '5'#').+& ($ ( $-+)$5 67 )/ 6$5 (' A8 /9$#$(+
(' .7$:/+ (' A
;2 U = {p(t) ∈ V ;p′(t) = 0} ,W = {p(t) ∈ V ;p(0) = p(1)} , V = P2(R).
Ex. 5.36 <'.'7# )$7 $& /++7(')$($& (+ ,'.+7 u = (−1, 8, 5) ∈ R3 '#
7'5$:/*$+ $ /$($ "#$ ($& %$&'& (' R
3
$%$ =+0
12 %$&' /$)>+) /$
32 {(0, 0, 1), (0, 1, 1), (1, 1, 1)}
;2 {(1, 2, 1), (0, 3, 2), (1, 1, 4)}
Ex. 5.37 <'.'7# )$7 $& /++7(')$($& (+ 6+5 )>+# + p(t) ∈ P3(R)8
($(+ 6+7 p(t) = 10 + t2 + 2t3, t ∈ R '# 7'5$:/*$+ $& &'-" ).'& %$&'&
(' P3(R)0
12 %$&' /$)>+) /$
32
{
1, 1 + t, 1 + t + t2, 1 + t + t2 + t3
}
;2
{
4 + t, 2, 2 − t2, t + t3
}
Ex. 5.38 <'.'7# )$7 $& /++7(')$($& (+ ,'.+7
(
2 5
−8 7
)
∈ M2 '#
7'5$:/*$+ $& &'-" ).'& %$&'& (' M20
 ! !"
#
$%&'( )* +!,-. /$0-1,
2
!( - ((3/-1!/!,
 ! "#$% &#'()'*&#
+!
{(
1 0
0 0
)
,
(
1 1
0 0
)
,
(
1 1
1 0
)
,
(
1 1
1 1
)}
Ex. 5.39 ,'&)'-.% /0# "#$% 1% M2 2/% &)'-%'3#
{(
1 0
1 0
)
,
(
1 1
0 0
)}
.
Ex. 5.40 4%.*52/% 2/% #$ &)).1%'#1#$ 1% p(x) ∈ Pn(R) &)0 .%6#7&8#)
9# "#$% B = {1, x, . . . , xn} :%


p(0)
p ′(0)
1
2!
p ′′(0)
!
!
!
1
n!
p(n)(0)


,
)'1% p(k)(0) .%;.%$%'-# # k<:%$*0# 1%.*=#1# 1% p %0 x = 0.
Ex. 5.41 >% {u1, . . . , un} :% /0# "#$% 1% V 0)$-.% 2/%
 ! {u1, u1 + u2, u1 + u2 + u3, . . . , u1 + · · · , un} :% /0 "#$% 1% V ;
+! $% αj 6= 0, j = 1, . . . , n %'-8#) {α1u1, . . . , αnun} :% /0# "#$% 1% V.
Caṕıtulo 6
Mudança de Base
6.1 Introdução, Exemplos e Propriedades
 
 ! "#! $ % &'&!() *+,- .$ / 01&%.1.$ 1& 2! &)&!&%3 1& 2!
&$(.4/ "&3 0#.) ( 1&! ".0#.0 52.%1 $& / %$#1&0.! 6.$&$ 1#$3#%3.$+
7 52& (.$$.0&! $ . &$321.0 .8 0. 9& / ! &$3. !21.%4/. / 00&: 2 $&;.:
/ ! 9& ( $$9<"&) &%/ %30.0 .$ / 01&%.1.$ 1& 2! "&3 0 / ! 0&).4/=. . 2!.
6.$& $.6&%1 >$& $2.$ / 01&%.1.$ / ! 0&).4/=. . 2!. 230.+
?&;. V 2! &$(.4/ "&3 0#.) @%#3.!&%3& 8&0.1 + ?&;.! B & C 6.$&$ 1&
V A 0!.1.$ (&) $ "&3 0&$ b1, . . . , bn & c1, . . . , cn, 0&$(&/3#".!&%3&+ B ! 
B 9& 2!. 6.$&: &'#$3&! αij ∈ R, 1 ≤ i, j ≤ n 3.#$ 52&
c1 = α11b1 + · · · + αn1bn
+
+
+
cn = α1nb1 + · · · + αnnbn.
C&$3. A 0!.: .$ / 01&%.1.$ 1& c1, . . . , cn, / ! 0&).4/=. D. 6.$& B $=. : 0&$>
(&/3#".!&%3&:
c1B =


α11
+
+
+
αn1

 , · · · , cnB =


α1n
+
+
+
αnn

 .
EF
 ! !"
#
$%&'( )* +&,!- .! ,/ 0!1/
"#$%&'() #)*+) &%,(-'+./0(#) )(1-# +) /((-2#%+2+) 2() 3#*(-#) 2+ 1+)# C
/(' -#4+./0+( 5+ 1+)# B %+ )#6$&%*# '+*-&7
MCB =


α11 · · · α1n
8
8
8
8
8
8
8
8
8
αn1 · · · αnn

 ,
/$9+) /(4$%+) )0+( ,(-'+2+) :#4+) /((-2#%+) 2# c1, . . . , cn /(' -#4+./0+( 5+
1+)# B. ; '+*-&7 MCB <# /=+'+2+ 2# '+*-&7 '$2+%./+ 2# 1+)# 2+ 1+)# B
:+-+ + 1+)# C.
;%*#) 2# '()*-+-'() + -#4+./0+( >$# #?&)*# #%*-# MCB # +) /((-2#%+2+)
2# $' 2+2( 3#*(- /(' -#4+./0+( 5+) 1+)#) B # C, 3#9+'() /('( :(2#'()
#%/(%*-+- + '+*-&7 2# '$2+%./+ 2# 1+)# #' $' #?#':4( %( R
3.
Exemplo 6.1 !"#$%&'& ( )(#& B %& R3 *!'+(%( ,&-!# .&/!'&# (1, 0, 1),
(1, 1, 1) & (1, 1, 2). !"#$%&'& /(+)0&+ ( )(#& C *!'+(%( ,&-!# .&/!'&#
(1, 0, 0), (0, 1, 0) & (0, 0, 1). 1"2!"/'& MCB.
@-#/&)+'() -#)(43#-
(1, 0, 0) = α11(1, 0, 1) + α21(1, 1, 1) + α31(1, 1, 2)
(0, 1, 0) = α12(1, 0, 1) + α22(1, 1, 1) + α32(1, 1, 2)
(0, 0, 1) = α13(1, 0, 1) + α23(1, 1, 1) + α33(1, 1, 2)
⇐⇒
(α11 + α21 + α31, α21 + α31, α11 + α21 + 2α31) = (1, 0, 0)
(α12 + α22 + α32, α22 + α32, α12 + α22 + 2α32) = (0, 1, 0)
(α13 + α23 + α33, α23 + α33, α13 + α23 + 2α33) = (0, 0, 1).
A' '('#%*( 2# -#B#?0+( %() :($:+-<+ $' :($/( 2# *-+1+4=( %#)*# :(%*(8
C(*# >$# /+2+ 4&%=+ +/&'+ -#:-#)#%*+ $' )&)*#'+ 2# *-D#) #>$+./0(#) /('
*-D#) &%/<(6%&*+) # >$# + '+*-&7 +))(/&+2+ + /+2+ $' 2#)*#) )&)*#'+) <# +
'#)'+8 E >$# '$2+ )0+( () %('#) 2+) 3+-&<+3#&) # ( )#6$%2( '#'1-(8
A*&4&7+%2( /('( 3+-&<+3#&) x, y # z, 1+)*+ -#)(43#-'() ( )#6$&%*# )&)*#'+


1 1 1
0 1 1
1 1 2




x
y
z

 =


a
b
c


 !"! #$%&'()*+
,
-'. /0/123'4 / 2&'2&#/(-(/4 !
"#$% a, b, c ∈ R. & '(')%*+ +,(*+ -% %./(0+1%#)% +


1 1 1
0 1 1
0 0 1




x
y
z

 =


a
b
c − a


,/2+ -/#(,+ '"1/3,4+" -% $+$+ 5"6 x = a − b, y = a + b − c % z = c − a.
7"*+#$" (a, b, c) = (1, 0, 0) "8)%*"' (α11, α21, α31) = (1, 1,−1).
7"*+#$" (a, b, c) = (0, 1, 0) "8)%*"' (α12, α22, α32) = (−1, 1, 0).
7"*+#$" (a, b, c) = (0, 0, 1) "8)%*"' (α13, α23, α33) = (0,−1, 1). 9%')+
:"6*+; "8)%*"'
MCB =


1 −1 0
1 1 −1
−1 0 1

 .
Exerćıcio 6.2 !" #$ %!&#'()!*$ +! *,*"-.! #(/"#0 *%(!%&1* MBC.
<%2+*"' +="6+ ,"*" +' ,""6$%#+$+' $% /* 0%)"6 '% 6%1+,("#+* ,"*
6%'5%()" + $/+' 8+'%' $% /* %'5+3," 0%)"6(+1 $% $(*%#'4+" >#()+?
@%2+* B % C 8+'%' $% /* %'5+3," 0%)"6(+1 $% $(*%#'4+" >#()+ V :"6*+$+';
6%'5%,)(0+*%#)%; 5%1"' 0%)"6%' b1, . . . , bn % c1, . . . , cn. 9+$" /* 0%)"6 v %*
V '%2+*
vB =


x1
?
?
?
xn

 % vC =


y1
?
?
?
yn


+' '/+' ,""6$%#+$+' ,"* 6%1+3,4+" A+' 8+'%' B % C, 6%'5%,)(0+*%#)%? @%
MCB = (αij) 6%56%'%#)+ + *+)6(B $% */$+#3,+ $+ 8+'% B 5+6+ 8+'% C, %#)4+"
,"*" cj =
∑n
i=1 αijbi, j = 1, . . . , n, "8)%*"'
v =
n∑
i=1
xibi =
n∑
j=1
yjcj =
n∑
j=1
yj
(
n∑
i=1
αijbi
)
=
n∑
i=1
(
n∑
j=1
αijyj
)
bi
 ! !"
#
$%&'( )* +&,!- .! ,/ 0!1/
"#$% #& '()*+,& +-(&)$&$% +#.%/*%,"0 & "/$%, $& 0",&1 2"," "0 .%*"/%0
b1, . . . , bn 03&" )1+14 0%-(%50% 6(% xi =
∑n
j=1 αijyj, i = 1, . . . , n. 7"/'%,4 %0*&0
'()*+,&0 n %6(&893"%0 :"$%, 0%/ %09/+*&0 #& 0%-(+#*% ;'"/,()& ,&*/+9+&)


α11 α12 · · · α1n
1
1
1
1
1
1
1
1
1
1
1
1
αn1 αn2 · · · αnn




y1
1
1
1
yn

 =


x1
1
1
1
xn

 ,
"( ,&+0 0+,:)%0,%#*%4
vB = M
C
BvC.
<%0(,+/%,"0 %0*% /%0()*&$" #& 0%-(+#*%
Proposição 6.3 !"#$ B ! C %#&!& '! ($ !&)#*+, -!.,/0#1 '! '0$!2&3#,
420.# V. ! vB ! vC /!)/!&!2.#$ #& +,,/'!2#'#& '! ($ '#', -!.,/
v ∈ V +,$ /!1#*+3#, 5#& %#&!& B ! C, /!&)!+.0-#$!2.! ! &! MCB 6! #
$#./07 '! $('#2*+# '! %#&! '# %#&! B )#/# # %#&! C !2.3#,
vB = M
C
BvC.
Exemplo 6.4 809#', θ ∈ R, +,2&0'!/! ,& -!.,/!&
u1 = (9"0 θ, 0%# θ) ! u2 = (− 0%# θ, 9"0 θ)
!$ R
2. :,&./! ;(! !&.!& -!.,/!& <,/$#$ ($# %#&!= B, '! R2 ! !2+,2./!
# $#./07 '! $('#2*+# '!&.# %#&! )#/# # %#&! C <,/$#'# )!1,& -!.,/!&
e1 = (1, 0) ! e2 = (0, 1). >2+,2./! #& +,,/'!2#'#& ', -!.,/ u = ae1+be2
+,$ /!1#*+3#, 5# %#&! B.
2"," & $+,%#03&" $% R
2
'% $"+0 =&0*& ,"0*/&/ 6(% u1 % u2 03&" )1+11 >%
α(9"0 θ, 0%# θ) + β(− 0%# θ, 9"0 θ) = (0, 0)
%#*3&" {
α 9"0 θ − β 0%# θ = 0
α 0%# θ + β 9"0 θ = 0
⇐⇒ α = β = 0,
 !"! #$%&'()*+
,
-'. /0/123'4 / 2&'2&#/(-(/4 !
"#$%
&'(
(
)#% θ − %'* θ
%'* θ )#% θ
)
= 1 6= 0.
+ ,-(.$/ MCB %'.0- &-&- "#. (αij), #*&'
(1, 0) = α11()#% θ, %'* θ) + α21(− %'* θ, )#% θ)
(0, 1) = α12()#% θ, %'* θ) + α22(− %'* θ, )#% θ),
12' 0' '12$3-4'*(' -
(1, 0) = (α11 )#% θ − α21 %'* θ, α11 %'* θ + α21 )#% θ)
(0, 1) = (α12 )#% θ − α22 %'* θ, α12 %'* θ + α22 )#% θ),
' )#,# 50- 3$%(# -*('%6 7-%(- .'%#43'. # %$%(',-
(
)#% θ − %'* θ
%'* θ )#% θ
)(
x
y
)
=
(
α
β
)
)25- %#428)9-# 0' &-&- "#.
(
x
y
)
=
(
)#% θ %'* θ
− %'* θ )#% θ
)(
α
β
)
=
(
α )#% θ + β %'* θ
β )#% θ − α %'* θ
)
.
:-/'*&# (α,β) = (1, 0) #7(',#% (α11,α21) = ()#% θ,− %'* θ).
;#4#)-*&# (α,β) = (0, 1), (',#% (α12, α22) = ( %'* θ, )#% θ). +%%$,6
MCB =
(
)#% θ %'* θ
− %'* θ )#% θ
)
.
+<#.-6 %' uB .'".'%'*(- -% )##.&'*-&-% &' u = ae1 + be2 )#, .'4-8)9-# =-
7-%' B ' uC -% )##.&'*-&-% &# ,'%,# 3'(#. )#, .'4-8)9-# =- 7-%' C, "'4-
".#"#%$8)9-# >? (',#%
uB = M
C
BuC =
(
)#% θ %'* θ
− %'* θ )#% θ
)(
a
b
)
=
(
a )#% θ + b %'* θ
b )#% θ − a %'* θ
)
.
 !"
#
$%&'( )* +&,!- .! ,/ 0!1/
Proposição 6.5 !"#$ B, C ! D %#&!& '! ($ !&)#*+, -!.,/0#1 n '02
$!3&0,3#14 5!$,&
MDB = M
C
BM
D
C.
Prova: !"#$% b1, . . . , bn &' (")&*"' +" B, c1, . . . , cn &' (")&*"' +" C "
d1, . . . , dn &' (")&*"' +" D. ,'$-+& $ -&)$./0$& M
C
B = (αij), M
D
C = (βij) "
MDB = (γij) ("%&' 12"
cj =
n∑
i=1
αijbi, dk =
n∑
j=1
βjkcj, dk =
n∑
i=1
γikbi. 3 4 5
6''7%8
dk =
n∑
j=1
βjkcj =
n∑
j=1
βjk
(
n∑
i=1
αijbi
)
=
n∑
i=1
(
n∑
j=1
αijβjk
)
bi,
/&%& b1, . . . , bn '0$& 94748 /&%:$*$-+& /&% $ ;29)7%$ "<:*"''0$& +" 4 8 &=>
)"%&'
γik =
n∑
j=1
αijβjk, 1 ≤ i, k ≤ n.
?"')$ $:"-$' 9"%=*$* 12" & 9$+& +7*"7)& +$ "<:*"''0$& $/7%$ *":*"'"-)$ &
"9"%"-)& +$ i>;"'7%$ 97-@$ " +$ k>;"'7%$ /&92-$ +$ %$)*7A MCBM
D
C. B&*)$-)&8
MDB = M
C
BM
D
C.
Proposição 6.7 !"#$ B C !"# # $ %$ #&"'() * +),-". / n /-0
$ 1#-)1". V. 21+3") " $"+,-4 MCB &)##%- -1* ,#" #+" -1* ,#" 5 /"/"
&), MBC, " $"+,-4 / $%/"1'(" /" !"# C &"," " !"# B.
Prova: !"# $%&$&'()*+#& #,-!%(&% -!.&' MCBM
B
C = M
B
B ! M
B
CM
C
B = M
C
C.
/!'-# .&'-%#% 01! MBB = M
C
C = I = (δij), &,2!
δij =
{
1 '! i = j
0 *#'& *&,-%3#%(&4
 !"! #$#%&
'
(&()* !
"# $ %$&'() (*#+&(*$*# *# ,'*#% n. "- ./$', 01# 2$3&$ %,3&'$' 01# MBB = I #
(3&, "# 2#% 3(%4/#35 4,(3 3# u1, . . . , un 36$, ,3 7#&,'#3 *$ 2$3# B #+&6$, M
B
B =
(αij) 3$&(38$) uj =
∑n
i=1 αijui, j = 1, . . . , n. 9'$5 .,%, u1, . . . , un 36$, /:(:5
4$'$ .$*$ j = 1, . . . , n, $ "1+(.$ 3,/1;.6$, *# .$*$ 1%$ *#3&$3 #01$;.6,#3 "#
*$*$ 4,'
αij =
{
1 3# i = j
0 .$3, .,+&'"$'(,5
,1 3#<$5 αij = δij.
Exerćıcio 6.8 !"#"$% & '()')*"+,-&) &,".& '&(& (%/&$%( ) %0%(,12,") 3454
6.2 Exerćıcios
Ex. 6.9 6)7*"8%(% &* 9&*%* B = {e1, e2, e3} % C = {g1, g2, g3} 8% :.
%*'&+,) ;%!)("&# V (%#&,")7&8&* 8& *%<:"7!% /)(.&



g1 = e1 + e2 − e3
g2 = 2e2 + 3e3
g3 = 3e1 + e3
=4 >%!%(."7% &* .&!("$%* .:8&7+,& 8& 9&*% B '&(& & 9&*% C? "*!)
1%? MCB? % 8& 9&*% C '&(& & 9&*% B? "*!) 1%? M
B
C.
54 @% & .&!("$ 8&* ,))(8%7&8&* 8) ;%!)( v %. (%#&+,-&) & 9&*% B?
"*!) 1%? vB? 1% 8&8& ')(


1
3
2

 %7,)7!(% & .&!("$ 8&* ,))(8%7&8&*
8% v %. (%#&+,-&) & 9&*% C? "*!) 1%? vC.
A4 @% & .&!("$ 8&* ,))(8%7&8&* 8) ;%!)( v %. (%#&+,-&) & 9&*% C? "*!)
1%? vC? 1% 8&8& ')(


2
3
−1

 %7,)7!(% & .&!("$ 8&* ,))(8%7&8&*
 ! !"
#
$%&'( )* +&,!- .! ,/ 0!1/
 ! v !" #!$%&'(%) % *%+! B, -+.) /!, vB.
Ex. 6.10 0)1+- !#! %+ *%+!+ )# !1% %+ B =
{
1, 1 + t, 1 + t2
}
! C ={
1, t, t2
}
 ! P2(R).
23 41')1.#! %+ "%.#-5!+ ! "6 %1&'% % *%+! B 7%#% % *%+! C, -+.)
/! MCB, ! % *%+! C 7%#% % *%+! B, -+.) /! M
B
C.
83 9! vB =


1
−4
6

 !1')1.#! vC.
:3 9! vC =


8
−1
3

 !1')1.#! vB.
;3 9! D =
{
1, t, t2
}
/! % *%+! '%1<)1-'% ! P2(R), !1')1.#! %+ "%=
.#-5!+ ! "6 %1&'% % *%+! B 7%#% % *%+! D ! % *%+! D 7%#% %
*%+! C, -+.) /!, MDB ! M
C
D, #!+7!'.->%"!1.!3
Ex. 6.11 0)1+- !#! ) +!?6-1.! +6*!+7%&') ! M2@
W =
{(
x y
z t
)
∈ M2; x − y − z = 0
}
.
23 A)+.#! B6!
B =
{(
1 1
0 0
)
,
(
1 0
1 0
)
,
(
0 0
0 1
)}
!
C =
{(
1 0
1 0
)
,
(
0 −1
1 0
)
,
(
0 0
0 1
)}
+(%) *%+!+ ! W.
 !"! #$#%&
'
(&()* !
 ! "#$%#&'( )* +)&',-(* .( +/.)#0$) .) 1)*( B 2)') ) 1)*( C ( .)
1)*( C 2)') ) 1)*( B3 ,*&% 4(3 MCB ( M
B
C3 '(*2($&,5)+(#&(!
6! "#$%#&'( /+) 1)*( D .( W3 &)7 8/( ) +)&',-
P =


1 1 0
0 0 2
0 3 1


*(9) ) +)&',- .( +/.)#0$) .) 1)*( D 2)') ) 1)*( B3 ,*&% 4(3
P = MBD.
 ! !"
#
$%&'( )* +&,!- .! ,/ 0!1/
Caṕıtulo 7
Exerćıcios Resolvidos – Uma
Revisão
 
 !" #$%&'"()* $%+ ! ,"$-*! (-$ !& +. / 0 +#&'#.*! + !*)1./*! 2(!3
#$,/* 4$5 + (- + !(-* /* 6( 1.-*! $"& $7*+$8
Ex. Resolvido 7.1 !"#$%&! '! V = {(x, y, z,w) ∈ R4; y = x, z = w2}
()* +' ),!"+-(.)!' &'&+#' /! R
4
0! &* !',+-() 1!2)"#+34
Resolução: 9*" 6( (0, 0, 1, 1) ∈ V -$! −1(0, 0, 1, 1) = (0, 0, −1, −1) 6∈
V. :!!.-; V ,<$* & (- !%$=#* 1 "*+.$)8 ¤
Ex. Resolvido 7.2 5!6+ A ∈ Mn &*+ *+2"#7 %&+/"+/+ /! )"/!* n.
 !"#$%&! '! W = {X ∈ Mn×1;AX = 0} 0! &* '&8!',+-() 1!2)"#+3 /!
Mn×1, ()* +' ),!"+-(.)!' &'&+#'4
Resolução:
>8 ? @$ O = (0) $ -$"+.5 n×1 ,()$8 A*-* AO = O, " -*! 6( O ∈ W.
B8 ? X, Y ∈ W λ ∈ R, ,"<$*; % )$! %+*%+. /$/ ! /$ !*-$ /$ -()".3
%).#$=#<$* %*+ !#$)$+ (!($.! ,"+ $! -$"+.5 ! ; "$-2& -; % )$! %+*3
%+. /$/ ! /* %+*/("* ,"+ -$"+.5 !; " -*!
A(X + λY) = AX + A(λY) = AX + λAY = O + λO = O.
C>
 ! !"
#
$%&'( )* +,+- 
#
$ $(. -+.('/$0(. 1 &2! -+/$.
3
!(
"#$%&'%# X + λY ∈ W.
(#')*+,-.#/ 0+1 W ,1 +. /+21/3&4)# 51%#$6&* 71 Mn×1. ¤
Ex. Resolvido 7.3 !"#!$%& # '()&'*+,"# -&$#%.+/ 0& P3(R) 1&%+0#
*#% S = {1, t, t2, 1 + t3}.
Resolução: 8#%1 0+1 t3 = (t3+1)−1. 9//6.: 7&7# p(t) = a0+a1t+a2t
2+
a3t
3 ∈ P3(R) 3#71.#/ 1/)$151$ p(t) = (a0−a3)+a1t+a2t2+a3(t3+1) ∈
[S]. ;#<#: P3(R) = [S]. ¤
Ex. Resolvido 7.4 !"#!$%& # '()&'*+,"# -&$#%.+/ 0& M2 1&%+0# *#%
S =
{(
0 1
0 0
)
,
(
0 0
−1 0
)}
Resolução: =1.#/ 0+1 A ∈ [S] /1 1 /#.1'%1 /1 1>6/%1. α,β ∈ R %&6/ 0+1
A = α
(
0 1
0 0
)
+ β
(
0 0
−1 0
)
=
(
0 α
−β 0
)
,
#+ /1?&: A ∈ [S] /1 1 /#.1'%1 /1 #/ 1*1.1'%#/ 7& 76&<#'&* 3$6')63&* 71 A
/@&# '+*#/A ¤
Ex. Resolvido 7.5 !"#!$%& (2 "#!3(!$# 4!.$# 0& 1&%+0#%&' *+%+
W = {X ∈ M3×1 : AX = 0},
#!0&
A =


0 1 0
2 1 0
1 1 4

 .
 !
Resolução:
X =


α
β
γ

 ∈ W ⇐⇒


0 1 0
2 1 0
1 1 4




α
β
γ

 =


0
0
0


⇐⇒


1 1 4
2 1 0
0 1 0




α
β
γ

 =


0
0
0

 ⇐⇒


1 1 4
0 −1 −4
0 1 0




α
β
γ

 =


0
0
0


⇐⇒


1 1 4
0 1 4
0 1 0




α
β
γ

 =


0
0
0

 ⇐⇒


1 1 4
0 1 4
0 0 −4




α
β
γ

 =


0
0
0


⇐⇒


1 1 4
0 1 4
0 0 1




α
β
γ

 =


0
0
0

 ⇐⇒ α = β = γ = 0,
"#$%&'%#(
W =





0
0
0





.
¤
Ex. Resolvido 7.6 !"#!$%& '( "#!)'!$# *!+$# ,& -&%.,#%&/ 0.%.
W = {X ∈ M4×1 : AX = 0},
#!,&
A =


1 1 −1 0
2 0 1 1
3 1 0 1
0 −2 3 1

 .
 ! !"
#
$%&'( )* +,+- 
#
$ $(. -+.('/$0(. 1 &2! -+/$.
3
!(
Resolução:
X =


α
β
γ
δ

 ∈ W ⇐⇒


1 1 −1 0
2 0 1 1
3 1 0 1
0 −2 3 1




α
β
γ
δ

 =


0
0
0
0


⇐⇒


1 1 −1 0
0 −2 3 1
0 −2 3 1
0 −2 3 1




α
β
γ
δ

 =


0
0
0
0


⇐⇒


1 1 −1 0
0 −2 3 1
0 0 0 0
0 0 0 0




α
β
γ
δ

 =


0
0
0
0


⇐⇒


1 1 −1 0
0 1 −3/2 −1/2
0 0 0 0
0 0 0 0




α
β
γ
δ

 =


0
0
0
0


⇐⇒


1 0 1/2 1/2
0 1 −3/2 −1/2
0 0 0 0
0 0 0 0




α
β
γ
δ

 =


0
0
0
0


⇐⇒
{
α = −γ/2 − δ/2
β = 3γ/2 + δ/2
,
"#$% &'(
X =


−γ/2 − δ/2
3γ/2 + δ/2
γ
δ

 = γ


−1/2
3/2
1
0

 + δ


−1/2
1/2
0
1

 ,
 !
"#$%&'%#(
W =




−1/2
3/2
1
0

 ,


−1/2
1/2
0
1



 .
¤
Ex. Resolvido 7.7 !"#!$%& '() *)+& ,# +'*&+-)."# /&$#%0)1 ,& R3
,),# -#% U = [(1, 0, 1), (1, 2, 0), (0, 2,−1)].
Resolução: !"#$"!% &%'%( (x, y, z) ∈ U )* * )#+*'%* )* *,-)%*+ α,β, γ ∈
R %&-) ./*
α(1, 0, 1) + β(1, 2, 0) + γ(0, 2, −1) = (x, y, z),
#/ )*0&( (x, y, z) ∈ U )* * )#+*'%* )* # )-)%*+& &1&-,# &2+-%* )#3/456&#


1 1 0
0 2 2
1 0 −1




α
β
γ

 =


x
y
z

 ⇐⇒


1 1 0
0 2 2
0 −1 −1




α
β
γ

 =


x
y
z − x


⇐⇒


1 1 0
0 1 1
0 −1 −1




α
β
γ

 =


x
y/2
z − x


⇐⇒


1 1 0
0