02_Espaços Vetoriais_UFMG

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+ y ;#(# &*7(3$7$2
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6-$ %$/!6$:$!- ~v = x~i+ y~j7 +.
;#+ x, y ∈ R7 0<- %0 ,-.1-9
3+36+0 =+ ~v +. $+2%><- ? /%0+
{~i,~j}'
34!'5,# 676787 @+A% ~v = 2~i + (−3)~j7 1-=+.-0 $+1$+0+36%$ 1-$
~v = (2,−3) ∈ R2' B+$,+/% ;#+7
2~i = (2, 0),
(−3)~j = (0,−3) + %00!.
~v = 2~i+ (−3)~j = (2 + 0, 0 + (−3)) = (2,−3)
 !" #$%&'(&() ) *+),&-.)/
#$%$& '()*+,*,-
C0 8+6-$+0 ~u = (x1, y1) + ~v = (x2, y2) 0<- !"#%!0 0+7 + 0-.+36+ 0+7
x1 = x2 + y1 = y27 + %00!.7 ~u = ~v'
34!'5,# 679727 C0 8+6-$+0 ~u = (2, 1) + ~v = (2, 1) 0<- !"#%!07
1-$D.7 ~p = (2, 1) + ~q = (1, 2) 3<- - 0<-'
#$%$# ./-0*12-3
@+A%. -0 8+6-$+0 ~u = (x1, y1) + ~v = (x2, y2) + λ ∈ R7 =+fi3+90+*
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#! λ~u = (λx1, λy1)!
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%
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A1. ~u+ ~v = ~v + ~u/
A2. (~u+ ~v) + ~w = ~u+ (~v + ~w)/
A3. ~u+~0 = ~u/
A4. ~u+ ~(−u) = ~00
' 1!"! #$!%&#$'" ~u ' ~v ' α, β ∈ R+ )',-&'
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P1 α(β~u) = (αβ)~u!
P2 (α + β)~u = α~u+ β~u!
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P4 1~v = ~v"
#$%&'()*+, " ! "#$%&
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<= )3&#.#($)2%.0 >)'%. ~u = (x1, y1), ~v = (x2, y2) # ~w = (x3, y3),
$#2%. 4*#?
• #2 @A1A,
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= (x2, y2) + (x1, y1) = ~v + ~u.
• #2 @A4A, -&#2%. .*3%& ~w = (a1, a2), $)+ 4*#
~0 = ~u+ ~w = (x1, y1) + (a1, a2) = (x1 + a1, y1 + a2)
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x1 + a1 = 0 # y1 + a2 = 0⇒ a1 = −x1 # a2 = −y1,
3%&$)($%, ~w = (−x1,−y1) = −~u0
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(α + β)~u = (α + β) · (x1, y1) = ((α + β)x1, (α + β)y1) .
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= (αx1, αy1) + (βx1, βy1)
" (''3#&
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= α(x1, y1) + β(x1, y1) = α~u+ β~u.
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)/=/ / ="'"06/*63#"0)/ =(' :2/:23"=(="' ="#/0')2(=('& ="3B(#/'
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"+,(84/
4(~u− ~v) +
1
3
~w = 2~u− ~w
:2")"0="#/' =")"2#30(2 / 6")/2 ~wC F(2( 3''/& D(2"#/' ,'/ =('
:2/:23"=(="' =(' /:"2(8G"' "0)2" 6")/2"'C H(8(#/'
4(~u− ~v) +
1
3
~w = 2~u− ~w ⇒
1
3
~w + ~w = 2~u− 4(~u− ~v)⇒
4
3
~w = −2~u+ 4~v
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4
3
~w = −2~u+ 4~v ⇒
4~w = −6~u+ 12~v ⇒
~w =
−6
4
~u+
12
4
~v ⇒
~w =
−3
2
~u+ 3~v.
()* !+$ ) ',-'".",./#$% 01 2)&$' )
~w =
−3
2
~u+ 3~v ⇒
~w =
−3
2
(3,−1) + 3(−1, 2)⇒
~w =
(
−9
2
,
3
2
)
+ (−3, 6)
$, 3$+ &$' '04 5 4 )''.&6 ~w =
(
−9
2
− 3,
3
2
+ 6
)
=
(
−15
2
,
15
2
)
%
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2
(1,−1)
8
3!fi+(56# 7822 9 !"#$ %!&'(')# *#$ %#'+ ,#("#+-8 :$!'.;
+ 4 $ 5 "$4
−−→
AB + $4.2 & !$ 3$!"$ A = (x1, y1) <"4 &.+)+ 
 & B = (x2, y2)8 :$&$ !) fi2,4) 9>8>>?% )' 0$$4+ !)+)' + 
−−→
AB
'#$ $-".+)' 3$4
−−→
AB = B −A% )''.&
−−→
AB = (x2 − x1, y2 − y1).
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$4. !")+$' AB% CD OP 4 34 ' !")& $ & '&$ 5 "$4 (3, 1)% 3$.'
−−→
AB = B −A = (1, 4)− (−2, 3) = (3, 1)
−−→
CD = D − C = (4, 3)− (1, 2) = (3, 1)
−−→
OP = P −O = (3, 1)− (0, 0) = (3, 1)
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~AB = B −A
 !"#$% &'&)(
~AB = (x2 −
x1, y2 − y1)
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3
1
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-/81/0$%$ a1, a2, a3 ∈ R3 0%7 A#-
~v = a1 ~v1 + a2 ~v2 + a3 ~v3 >&'F?
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<!'6(' +"6$. {~v1, ~v2, ~v3}=>
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(,"5$ !"6 fi%*,"6 <J>JG=. <J>JK= ' <J>JL=>
+),&'() J* +"5" -$!($ P !$ '6-"4$ <R3= +$,,'6-$!5' *&" (',!"
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 !"#$% &'&() *+%,-
xy
 !"#$% &'&.) *+%,-
xz
 !"#$% &'&/) *+%,-
yz
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2345$6%,7- % fi"#$% 9&'&1:; <5=-4)
A = (2, 0, 0) > *-,<- ,- 5!?- 7-4 x @#%,7- y = z = 0'
B = (0, 4, 0) > *-,<- ,- 5!?- 7-4 y @#%,7- x = z = 0'
C = (0, 0, 3) > *-,<- ,- 5!?- 7-4 z @#%,7- x = y = 0'
D = (2, 4, 0) > *-,<- ,- *+%,- xy @#%,7- z = 0'
E = (0, 4, 3) > *-,<- ,- *+%,- yz @#%,7- x = 0'
F = (2, 0, 3) > *-,<- ,- *+%,- xz @#%,7- y = 0'
A5B% ~v = x~i + y~j + z~k; 5= @#5 x, y 5 z 4C- %4 D-=*-,5,<54 75
~v ,% 3%45 D%,E,!D% {~i,~j,~k}; D-=- fiF5=-4 *%$% 65<-$54 ,- *+%,-'
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2
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 !"#$% &'(+* ~v = 2~i+ 4~j + 3~k
, -./0#/1. 2.$3%4. 5.$ #3 5./1. 6 5.$ #3% 7%86 -./81!1#!
#3 8!8163% $626$6/-!%9'
• , 685%:. 163 1$;8 4!36/8<68= .# 860%= > "$(3('!+%(#+),=
5.$?#6 ?#%9?#6$ #3% 46 8#%8 7%868 163 1$;8 @61.$68'
• , 59%/. 163 4!36/8A. &= .# 860%= 4(3('!+%(#+),'
• B $61% 163 4!36/8A. += .# 860%= 5+(3('!+%(#+),'
C.$ .#1$. 9%4.= % $65$686/1%:A. "6.3>1$!-% 4. -./0#/1. R > % $61%
-D%3%4% 46 $!") $!),' , 5$.4#1. -%$168!%/. R×R = R2 E.# %!/4%=
R
2 = {(x, y);x, y ∈ R}F 163 -.3. $65$686/1%:A. "6.3>1$!-% .
6,)+# .)$"!%()+#' G 5.$ fi3= . 5$.4#1. -%$168!%/. R×R×R = R3
E.# %!/4%= R
3 = {(x, y, z);x, y, z ∈ R}F 163 -.3. $65$686/1%:A.
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#$%$& '()*+,*,- - ./-0*12-3
8!fi+(7:# ;<2; E !"#$%#%&'< I.!8 @61.$68 ~u = (x1, y1, z1) 6 ~v =
(x2, y2, z2) 8A. !"#%!8 86= 6 8.36/16 86= x1 = x2= y1 = y2 6 z1 = z2'
39
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 !"#$% &'(&) *+,%
$+%-. R'
 !"#$% &'(() /-%01
2%$,+3!%01. R
2 =
R× R'
 !"#$% &'(4) 536%71
2%$,+3!%01. R
3 =
R× R× R'
,(fi.+%/& 0123 8 !"# $ %&!'()! *!& (" $+,#-#&.1 9%:13
13 ;+,1$+3 ~u = (x1, y1, z1) + ~v = (x2, y2, z2) + λ ∈ R. :+fi0+=3+)
• ~u+ ~v = (x1 + x2, y1 + y2, z1 + z2)
• λ~u = (λx1, λy1, λz1)
>% 31?% 2~i + 4~j + 3~k. 3%@+0:1
A#+
2~i = 2(1, 0, 0) = (2, 0, 0)
4~j = 4(0, 1, 0) = (0, 4, 0)
3~k = 3(0, 0, 1) = (0, 0, 3)
B@3+$;%0:1 01 6-%01 xy. ,+?13
A#+)
~v = 2~i+ 4~j + ~k
= (2, 0, 0) + (0, 4, 0)︸ ︷︷ ︸+(0, 0, 3)
 !"#$ !"#$%" &
= (2, 4, 0) + (0, 0, 3)= (2, 4, 3)
 !"#$% &'(C) D 31?% :13 ;+=
,1$+3 A#+ +3,E1 01 6-%01 xy.
2~i+ 4~j. F !-#3,$%:% 6+-1 ;+,1$
 !"#$%"&'. +0A#%0,1 % 31?% :1
;+,1$ !"#$%"&' %1 ;+,1$ 3~k $+=
3#-,% 01 ;+,1$ ~v '
40
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
2
 !" 
3!fi+(56# 7829 !"#$ %!&'(')# *#$ %#'+ ,#("#+-8 !" A =
(x1, y1, z1) " B = (x2, y2, z2) #$% &%'# (%)*%# +,-'#+,". )% "#(-/%0
")*$%
−−→
AB = (x2 − x1, y2 − y1, z2 − z1)
 !"#$%! 1& 23 4"5 &" "#6."4". ~v = 2~i+3~j+4~k0 (%&"3%# "#6."4".
~v = (2, 3, 4)7 8##'30
~i−~j = (1,−1, 0)
2~i− 3~j + ~k = (2,−3, 1)
4~k = (0, 0, 4)
"3 (-.*'6,9-.0 {~i,~j,~k} = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}7
3!fi+(56# 782: .#()'/0# )! ,1$12!2'+3# )! %#'+ !"#4
$!+-8 !" &%'# 4"*%."# ~u = (x1, y1, z1) " ~v = (x2, y2, z2) #$% 6%9')":
-."# %, (-.-9"9%#;0 "<'#*" ,3 )=3".% λ ∈ R0 *-9 +," ~u = λ~v0 %,
#">- (x1, y1, z1) = λ(x2, y2, z2)7 2#*- ? - 6%)&'/$% &" (-.-9"9'#3% &"
&%'# 4"*%."#7 @"(."#")*-3%# (%. ~u//~v &%'# 4"*%."# ~u " ~v (-.-9"9%#7
;<!'=,# 789828 A"*".3')-. %# 4-9%."# &" m " n (-.- +," #">-3
(-.-9"9%# %# 4"*%."# ~u = (m+ 1, 3, 2) " ~v = (2, 1, 2n)7
B-.- ")6%)*.-.3%# m " n0 '."3%# ,#-. - 6%)&'/$% &" (-.-9":
9'#3% &" &%'# 4"*%."#0 -##'30 *"3%#
(m+ 1, 3, 2) = λ(2, 1, 2n),
%, #">-0 

m+ 1 = 2λ
3 = λ
2 = 2nλ
.
C +," ."#,9*- "3 m = 5 " n = 1/37
41
 ! "!#$%&! '()&*+$+!
 !" #$%&'(
 !"#$ $%&$' $()!*+!,-" .%! %, /!#-) (-+! "!) +!0-,(-"#- "-1
%,$ 0-,12*$34- &2*!$) +! -%#)-" /!#-)!"5 6-*7!0!,-" - 0-*0!2#-
+! 1$"! -)#-*-),$& ! $()!*+!,-" .%! 8 (-""9/!& %":;&- ($)$ +!"0)!;
/!) .%$&.%!) /!#-) *%, (&$*- <-% !"($3-= 0--)+!*$+-' 0-,- %,$
0-,12*$34- &2*!$) +-" /!#-)!" +!"#$ 1$"!5 >&8, +2""-' #$,18,
/!)2fi0$,-" $&@%,$" ()-()2!+$+!" +$" -(!)$3A!" !*#)! /!#-)!"5
 !) *+,-,./.$%
B5 CD()2,$ - /!#-) ~w = (1, 1) 0-,- 0-,12*$34- &2*!$) +! ~u =
(−2, 1) ! ~v = (1,−1)5
E5 F%$2" "4- $" 0-*+23A!" +! a ! b' *G,!)-" )!$2"' ($)$ .%! -"
/!#-)!" +- (&$*- ~u = (2a+1, 1) ! ~v = (3, 2b−3) "!H$, 2@%$2"I
J5 K$+-" -" /!#-)!" ~u = (2,−4)' ~v = (−5, 1) ! ~w = (−12, 6)'
+!#!),2*$) k1 ! k2' #$& .%! ~w = k1~u+ k2~v5
L5 C*0-*#)! -" *G,!)-" λ1 ! λ2' #$& .%! ~w = λ1~v1 +λ2~v2' "!*+-
~v1 = (1,−2, 1)' ~v2 = (2, 0, 4) ! ~w = (−4,−4,−10)5
M5 - .%$+)$+- ABCD #!,;"! A = (−1,−3) ! B = (5, 6)5
F%$2" "4- $" 0--)+!*$+$" +-" /8)#20!" C ! DI
42