algebra lineal unad 2016

8/18/2019 algebra lineal unad 2016

1/30

=

:

=. 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

\


2/30

'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' :=


3/30

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' :=


4/30

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' :=


5/30

'O*+' 'O-R /*+' W6' :=

;


6/30

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


7/30

'O*+' 'O-R /*+' W6' :=

'6*

&6*

&66


8/30

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


9/30

4


10/30

3J7

38

-gth`h irmbh

'O*+' 'O-R /*+' W6' :=

7 "hrcig 4;


11/30

:;

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' :=


12/30

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


13/30

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' :=


14/30

'O*+' 'O-R /*+' W6' :=


15/30

(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.


16/30

'O*+' 'O-R /*+' W6' :=


17/30

'O*+' 'O-R /*+' W6' :=


18/30

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' :=


19/30

$. 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' :=


20/30

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' :=


21/30

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' :=


22/30

/ 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


23/30

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' :=


24/30

= 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

'O*+' 'O-R /*+' W6' :=


25/30

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' :=


26/30

: : =

= : :

> 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' :=


27/30

>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&=)


28/30

' 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:


29/30

: : =

= : :

> 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


30/30

' 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 )

algebra lineal unad 2016

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