algebra lineal unad 2016
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=
:
=. Wem dhsbm sg pmrm ge om pmrg` `g ue bumrth. Om gsqucem cengrchr
czqucgr`m `g om pmrg` sg sgogbbchem bhdh go hrcige `g ue scstgdm `g
bhhr`gem`ms bmrtgscmems ge `hs `cdgeschegs. Rc om dhsbm gstí pmrm`m
ge go pueth qug tcgeg bhhr`gem`ms (:, =) d, (m) ¸quã tme oghs gstí `g om
gsqucem `go bumrth! (") ¸#uío gs su phscbc$e ge bhhr`gem`ms phomrgs!
Rcge`h (:,=) go pueth `g u"cbmbc$e `g om dhsbm % `gsbrc"cge`h ue trcmeiuoh
rgbtíeiuoh ph`gdhs `g&ecr.
'O*+' 'O-R /*+' W6' :=
:,=
6,6
\
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'pocbme`h /ctíihrms.
::2=:0 -:
:¸(¸:¸+=:)4 \
¸∛ ̧
-0 :.:7
(m) ¸quã tme ogkhs gstí `g om gsqucem `go bumrth8 -0 :.:7
0r
Vmrm om bhhr`gem`m phomr vmdhs m gebhetrmr go íeiuoh qug
sg igegrm
'O*+' 'O-R /*+' W6' :=
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Vhr trcihehdgtrm
tgegdhs
-0 r#hs μ > 0 rRe μ tgecge`h go 9mohr ge - % go 9mohr ge >
prhbg`gdhs
-0 r#hs μ < 0 bhs√= x
r < 0bhs
√= :
:.:2>
< 0 bhs√=
6.)
Rm"cge`h qug eh bhehbgdhs go íeiuoh % egbgsctmdhs go íeiuoh
pmrm `gtgrdcemr om bhhr`gem`m phomr, prhbg`gdhs m rgmoczmr om sciucgetg
nhrduom.
1me μ 0S \
tmeμ4=:
μ4 tme√= =
:
μ4 tme√= (6.7) μ4:>.7>¶
'O*+' 'O-R /*+' W6' :=
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6
R
+
7d
7;
6
R
+
3d6
R
+
7d6
R
+
/0 (:,=) /0 :.:7 :.3 0 : c 2
=
(!) ¸"uío gs su phscbc#e ge bhhr`gem`ms phomrgs8
/0 (:,=) V$ :%:&' :'%' $
(r, ) $ (:%:&', :'%'*)
:. We muth sg `gspomzm 7 d `go 6hrtg 7; mo stg, ougih 3 d
`go Rur ; mo stg % &emodgetg 7 d mo Rur. ?moomr om `cstmebcm %
`crgbbc$e m om qug qug`h `go pueth `g cecbch ge nhrdm moig"rmcbm %
irí&bm
'O*+' 'O-R /*+' W6' :=
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'O*+' 'O-R /*+' W6' :=
;
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We muth sg `gspomzm 7 d `go 6hrtg
7; mo stg, ougih 3 d `go Rur ; mo stg %
&emodgetg 7 d mo Rur
V+ -./++ 207>"+ +
"+-V+909/0%
#hhr`gem`ms /homrgs
#hhr`gem`m *gbtmeiuomr
7 7 6 :< c 3 =6k
3 R 7 :6 c 3 ;&6k
7 R " 6 c 3 &66k
*4 0 '4 2 54 2 #4
*% 0 '% 2 5% 2 #%
Z4∛ Zx:
+ Zy:
μ4tme√= Zy
Zx
'O*+' 'O-R /*+' W6' :=
?moomr om
`cstmebcm %
`crgbbc$e m om
qug qug`h `go
pueth `g cecbch
ge nhrdm
moig"rmcbm %
irí&bm
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'O*+' 'O-R /*+' W6' :=
'6*
&6*
&66
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7
7
-
> 0
3
-
> 0
V>"9+ V>/12+?
'4 0 :38 '% 0 =3
54 0 :3 5%0 F77(M)
'O*+' 'O-R /*+' W6' :=
'% 0 ' sce 7 sce 70 .3
'%0 7 (.3) ' %0=3
Mx4 M bhs 26 bhs70 .J
Mx4¸ 7(.J) '40 :38
5% 0 ' sce sce 0
.J
5%0 3 (.J) 5%0 F7 (M)
Jx4Jbhs >6 bhs0 .3
#40 #%07 (M)
7
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4
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3J7
38
-gth`h irmbh
'O*+' 'O-R /*+' W6' :=
7 "hrcig 4;
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:;
F.=7 '
3.: 5
3.8F #
::3;
7. Wem pmrtGbuom g4pgrcdgetm trgs `gspomzmdcgeths subgsc9hs ge ue pomeh,
bhdh sciugN F.=7 d R+, 3.: d , % 3.8F d ge uem `crgbbc$e `g F; 6. ocm go
gg 4 mpuetme`h mo gstg % go gg % mpuetme`h Imbcm go ehrtg, % Imoog (m) oms
bhdphegetgs `g bm`m `gspomzmdcgeth, (") oms bhdphegetgs `go
`gspomzmdcgeth rgsuotmetg, (b) om dmiectu` % `crgbbc$e `go `gspomzmdcgeth
rgsuotmetg, % (`) go `gspomzmdcgeth qug sg rgqugrcrí pmrm trmgr `g eug9h m om
pmrtGbuom Imstm go pueth `go mrrmequg.
#hhr`gem`ms /homrgs #hhr`gem`m
*gbtmeiuomr
F.=7 ::3 R+ 4 :%5: c 4
:%5:k
'O*+' 'O-R /*+' W6' :=
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F.=7
::3
M :.8:
M :.8:
3.: 6 7 %:' c 3 6k
3.8F : 6 " %&; c 3:%'6k
?moog (m) oms bhdphegetgs `g bm`m `gspomzmdcgeth
Vmrm tgegdhs
' 0 F.=7 d
'- 0 M :.8=88
'> 0 M :.8=88
< 0 ::3
Vmrm 7 tgegdhs
50 3,: d
5- 0 3,:
5> 0
< 0
Vmrm " tgegdhs
# 0 3,8F d
#4 03.7F
'O*+' 'O-R /*+' W6' :=
My 4 M sce ::7 sce ::74 -6.969
My4 0.=2 (-6.969) M y4 - :.3=33
Mx4 M bhs ::7 bhs ::74 - 6.969
Mx4¸ 0.=2 (-6.969) Mx4 -:.3=33
Jx 4 7.:>
7.:> c + 6k
#% 0 ' sce : sce : 0 .F7J7
#%0 3,8F (.F7J7) # % 0 :.
Bx4 M bhs:> bhs:0 .J8JJ
Bx4¸ 3,8F(.J8JJ) #40 3.7F
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3,8F
:
3,7F
:,
B.J
M:.7J;
B.J
M .7:
#%0:.
< 0:
(") oms bhdphegetgs `go `gspomzmdcgeth rgsuotmetg
*4 0 '4 2 54 2 #4
*4 0 M:.8: 2 3.: 2 3.7F 0 B.J
*% 0 '% 2 5% 2 #%
*% 0 M:.8: 2 2 :. 0 M.7:
B.J c M .7:
Z4∛ Zx:+ Zy: 0 ∛ (9.>< 0
tme√=(√6.60=>>) 0 M:.7J;
Z49.>< M:.7J;
(B.J, M:.7J;)
'O*+' 'O-R /*+' W6' :=
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'O*+' 'O-R /*+' W6' :=
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(b) om dmiectu` % `crgbbc$e `go `gspomzmdcgeth rgsuotmetg
'O*+' 'O-R /*+' W6' :=
M :, 7J; 46%&:B.J
F%'<
Lmiectu` 0 B.J d
crgbbc$e 0 :.7J; surgstg.
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'O*+' 'O-R /*+' W6' :=
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'O*+' 'O-R /*+' W6' :=
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F. m`hs ohs 9gbthrgsN
u 4 -c + :k -0a
w 4 :c-2k+a
v4 -0c+2k+:a
B!bu!"
. u . w# w . v
'O*+' 'O-R /*+' W6' :=
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$. u x v # u x w
b. (u x w ). %
&. Bhs ( u# w)
3. We Icpgrdgrbm`h qucgrg hngrtmr trgs bomsgs `g "me`gmsN ', 5 % #.
Om "me`gm ' bhetcgeg F i `g qugsh dmebIgih, = i `g rhqugnhrt %J i `g bmdgd"grtC
om "me`gm 5 bhetcgeg =: i `g bm`m ueh `g ohs trgs tcphs `g qugshmetgrchrgsC
% om "me`gm #, bhetcgeg =3 i `g qugsh dmebIgih, J i `g rhqugnhrt %J i `g bmdgd"grt.
Rc sg qucgrg smbmr m om 9getm
3 "me`gms `go tcph ',
J `g 5 %
= `g #,
'O*+' 'O-R /*+' W6' :=
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F i = i J i
=: i =: i =: i
=3 i J i J i
7-7 7-=
' 5 #
FK3 2 =:KJ 2 =3K= 0 :
=K3 2 =:KJ 2 JK= 0 :3
JK3 2 =:KJ 2 JK= 0 :=
L
*
#'
+"tãe dmtrcbcmodgetg om bmetc`m` qug egbgsctmríe, ge Acohirmdhs `gbm`m uem `g oms trgs bomsgs `g qugshs.
5me`gm '
5me`gm 5
5me`gm '
L'1*#'O L61
'O*+' 'O-R /*+' W6' :=
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6 O+D*'L+R6 D*'L+R
L
*
#'
06 =:6 =76
=>6 =:6 66
=
=666 0
:>.>
:7.>
:=.>
L 0 dmebIgih
* 0 rhqugnhrt
#' 0 bmdgd"grt
3.= 1rgs pgrshems, ', 5, #, qucgrge bhdprmr oms sciucgetgs bmetc`m`gs `g
nrutmN
'N : Ai `g pgrms, = Ai `g dmezmems % Ai `g emrmems.
5N : Ai `g pgrms, : Ai `g dmezmems % F Ai `g emrmems.
#N = Ai `g pgrms, : Ai `g dmezmems % 7 Ai `g emrmems.
e go pug"oh ge go qug 9c9ge Im% `hs nrutgrGms E= % E:. e E=, oms pgrms
bugstme =.3 gurhsH Ai, oms dmezmems = gurhH Ai, % oms emrmems :
'O*+' 'O-R /*+' W6' :=
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/ L 6
'
5
#
7. 2 .J 2=:
7. 2 =. 2 J
=.J 2 =. 2
7 2 = 2 =:
7 2 : 2 J
=.3 2 : 2
' 5 #
/
L
6
gurhsHAi. e E:, oms pgrms bugstme =.J gurhsHAi, oms dmezmems ,J
gurhsHAi, % oms emrmems : gurhs H Ai.
: = >
: : 0
= : 2
=.7 =.<
= 6.<
: :
: = >
: : 0
= : 2 x
=.7 =.<
= 6.<
: :0 0
=> =>.0
=2 =2.:
3.7 3.0
`) ?moomr om ce9grsm `g om dmtrcz `he`g sg rgprgsget$ om bmetc`m` `g nrutm
(pgrms, dmezmems % emrmems) qug qucgrg bhdprmr bm`m pgrshem (', 5,
#).
: : =
= : :
> 0 2
'O*+' 'O-R /*+' W6' :=
/EE
6
L
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b) phr Dmuss Vhr`íe % ougih phr `gtgrdcemetgs utcoczme`h om n$rduom '
M=0 K '`'
e9grsm phr Dmuss Vhr`íe
: : =
= : :
> 0 2
= 6 6
6 = 6
6 6 =
E=M= 0 n= @ n:
E= : : = =
E: M= M: M: M M=
E=M= = M= = M=
= 6 √== : :
> 0 2
= √= 66 = 6
6 6 =
E:M=
0 n: @ n=
E: = : : =
ME= M= M = M= =
E:M= : 7 M= :
= 6 √=
6 : 2
> 0 2
= √= 6
√= : 66 6 =
E7M= 0 n7 @ n=
ME= M M
E7 F 7 =
E7M= F 8 M =
= 6 √=6 : 2
6 0 3
= √= 6√= : 6√> > =
E:M= 0=
: n:
=
:
n:
=2
:
√=:
=
E:M= = √=: =
'O*+' 'O-R /*+' W6' :=
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= 6 √=
6 = 2
:
6 0 3
= √= 6√=
:= 6
√> > 6
E7M= 0 n 2√0 n :
MFn: MF M : √ =
E7 F 8 M
E7M= 7 MF : =
= 6 √=
6 = 2
:
6 6 2
= √= 6√=
:= 6
√0 : =
E7M= 0=
2 n7
=
2
n7
=√0
2
:
2
=
2
E7M= =√0
2
:
2
=
2
= 6 √=
6 = 2
:
6 6 =
= √= 6√=
:= 6
√02
:
2
=
2
E:M= 0 n: @2
: n7
E: =2
:
√:
=
√2:
n7
√
: √
√:
E:M= = 6 √
:
= 6 √=6 = 6
6 6 =
= √= 6
2
:6 √=
:
√0
2
:
2
=
2
E=M= 0 n= 2 n7
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E= = M= = M=
n7 =√
2
:
2
E=M= = √2 √2
= 6 6
6 = 6
6 6 =
√=
2
√=
2
=
2
2
:6
√=
:
√0
2
:
2
=
2
(= 6 6
6 = 6
6 6 =
|
√=
2
√=
2
=
2
2
:6
√=
:
√0
2
:
2
=
2 )
/hr Dmuss Vhr`me
' 0
(: : =
= : :
> 0 2
)
'M=0
√=2
√=2
=
2
2
:6
√=:
√02
:
2
=
2
"hdprh!me`h
' K 'M=0
'O*+' 'O-R /*+' W6' :=
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: : =
= : :
> 0 2
√=2
√=2
=
2
2
:6
√=:
√0
2
:
2
=
2
0
= 6 6
6 = 6
6 6 =
m== m=: m=2
m:= m:: m:2
m2= m2: m22
== 0:∙√=
2+ :∙2
:+ =∙√0
24¸
√:2+2+
√024=
':= 0=∙√=
2+
:∙2:
+:∙√0
24¸
√=
2+2+
√<
246
'7= 0>∙√=
2+
0∙2:
+2∙√0
246
'=: 0:∙√=
2+:∙6+
=∙:2
46
:: 0=∙√=
2+:∙6+
:∙:2
4=
' 7: 0>
∙√=
2+0∙6+
2
∙:
246
'=7 0:∙=
2+
:∙√=:
+=∙=
246
'7: 0
=∙=
2
+:∙√=
:
+:∙=
2
46
&&0>∙=
2+
0∙√=:
+2∙=
24=
= 6 6
6 = 6
6 6 =
m== m=: m=2
m:= m:: m:2
m2= m2: m22
Rg mbm"m `g bhdprh"mr qugduotcpocbme`h om dmtrcz hrcicemophr om ce9grsm sg h"tcgeg omdmtrcz c`getc`m`.
'O*+' 'O-R /*+' W6' :=
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>evgrsm phr `gtgrdcemetgs%
' 0 (: : == : :> 0 2
)
'M= 0 ce9grsm
=
⃒M ⃒0 ce9grsh `gtgrdcemetg
('K) t 0 dmtrcz trmespugstm `g om m`uetm
'K 0 dmtrcz m`uetm
=% bmobuomdhs go `gtgrdcemetg
m== m=: m=2
m:= m:: m:2
m2= m2: m22
m== m=: m=2
m:= m:: m:2
m2= m2: m22
m== m=: m=2
m:= m:: m:2
m2= m2: m22
m== m=: m=2
m:= m:: m:2
m2= m2: m22
$ (m== (m:: Mm&&) 3 m=: (m:&Mm&=) 3 m=& (m:=Mm&:)) 4 m==(m:&Mm&:) J m=:(m:=Mm&&)J m=& (m::Mm&=)
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' 0 (: : == : :> 0 2
)
: : =
= : :
> 0 2
: : =
= : :
> 0 2
(m== (m:: Mm&&) 3 m=: (m:&Mm&=) 3
m=& (m:=Mm&:))
(:K:K7) 2 (:K:K) 2 ( =K=KF) 0 =:2:F 2F 0 F
: : =
= : :
> 0 2
: : =
= : :
> 0 2
4 m== (m:&Mm&:) J m=: (m:=Mm&&) J
m=& (m::Mm&=)
M (: K:KF) M (:K=K7) @ (=K:K) 0 M= @
@ =: 0 M 7F
$ ;6 4&; $ '
>evgrsh ̀ gtgrdcemetg
=
⃒M ⃒4=
>
:. bmobuomdhs om dmtrcz
m`kuetm.
'! &kueh & M# &eh&h *h" &k M# s !
"s*us & ! "c, & bhbh"s & M
' 0
: : =
= : :
> 0 2
bhbh"s
: : == : :
> 0 2
m== 0 |: :
0 2| 0:K7 @ :KF 0 MJ 0 M:
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: : =
= : :
> 0 2
m=:0 √|= :> 2| 0M(=K7 @ K:) 0 M72=:0 8
: : =
= : :
> 0 2 m=70 |= :> 0| 0
=KF @ K: 0 FM=: 0MJ
: : =
= : :
> 0 2 m:=0 √|: =0 2| 0 M
(:K7 @ FK= )0 M2F0 M:
: : == : :
> 0 2
m::0 |: => 2| 0:K7 @ =K 0 M 0
: : =
= : :
> 0 2 m:70 M |: :> 0| 0
M(:KF @ K:)0 MJ2=: 0 F
: : =
= : :
> 0 2 m7=0 |: =: :| 0
:K: @ :K= 0 FM: 0 :
: : =
= : :
> 0 2 m7:0 √|: == :| 0 M
(:K: @ =K=) 0 MF2= 0 M7
: : == : :
> 0 2 m770 |: := :| 0
:K: @ :K= 0 FM: 0 :
m== m=: m=2
m:= m:: m:2
m2= m2: m22
'== 0 |m:: m:2m2: m22|
'=: 0 |m:= m:2m2= m22|
'=70 |m:= m::m2= m2:|
':= 0 |m=: m=2m2: m22|
':: 0 |m== m=2m2= m22|
':7 0 |m== m=:
m2= m2:
|
'7= 0 |m=: m=2m:: m:2|
'7: 0 |m== m=2m:= m:2|
'77 0 |m== m=:m:= m::|
#hnmbthrgs
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' 0
m== m=: m=2
m:= m:: m:2
m2= m2: m22
`k $
m== m:= m2=
m=: m:: m2:
m=2 m:2 m22
' 0
(√: 3 √<
√: 6 0: √2 : )
'` ' 0 (√: √: :3 6 √2√< 0 :
)
-mtrcB cevgrsm phr`gtgrdcemetgs
=
⃒M ⃒4=
>
'M= 0=
> (√: √: :3 6 √2√< 0 : )
'M= 0 (√:
>
√:
>
:
>
3
>
6
>
√2
>
√<
>
0
>
:
> )'M= 0
(√=
2
√=:
=
2
2
:6 √=
:
√02
:
2
=
2 )
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