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Zbiór zadań z algebry liniowej - Maria Ekes, Jacek Kłopotowski - ebook – Ibuk.pl

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481 Z|nd}df/ }h mhvol D mhvw grgdwqlr rnuhvorqd/ wr lvwqlhmh pdflhu} rgzurwqd D/ nwöud wh } mhvw grgdwqlr rnuhvorqd1

491 Xgrzrgqlf/ }h pdflhu} D mhvw qlhxmhpqlh rnuhvorqd zwhg| l w|onr zwhg|/

jg| lvwqlhmh pdflhu} E wdnd/ }h D @ EAE1

4:1 Z|nd}df/ }h pdflhu} D @ EAE/ jg}lh E mhvw pdflhu} d nz dgudwrz d/ mhvw grgdwqlr rnuhvorqd zwhg| l w|onr zwhg|/ jg| ghw E 9@ 31

4;1 Qlhfk D e hg}lh pdflhu} d qlhxmhpqlh +qlhgrgdwqlr, rnuhvorq d1

d, Z|nd}df/ }h god nd }ghjr { 5 U?} zduxqnx {AD{ @ 3 z|qlnd/ l } D{ @ 31 e, Z|nd}df/ }h mhvol D mhvw qlhrvreolzd/ wr mhvw grgdwqlr +xmhpqlh, rnuhvorqd1 4<1 Xgrzrgqlf/ }h uhodfmd uözqrzd }qrvfl irup nzdgudwrz|fk mhvw }zurwqd/

v|phwu|f}qd l su}hfkrgqld1

531 Xgrzrgqlf/ }h nd }gd irupd nzdgudwrzd i r v|jqdwxu}h +s> t, mhvw uözqr0 zd }qd iruplh ndqrqlf}qhm

k+|, @ SR

'|2  Rn^S

'Rn|2=

541 Xgrzrgqlf/ }h irupd nzdgudwrzd i = U? $ U mhvw qlhxmhpqlh +qlhgr0 gdwqlr, rnuhvorqd zwhg| l w|onr zwhg|/ jg| }elöu i{ 5 U?= i +{, @ 3j mhvw srg0 su}hvwu}hql d olqlrz d su}hvwu}hql U?1

551 ]edgdf rnuhvorqrvf irup nzdgudwrz|fk=

d, i = U $ U/ i+{, @ {2. {22. {2  5{{2. {2{ >

e, i = U $ U/ i+{, @ {2 5{22 {2  {{ . 5{{21

561 ]edgdf/ z}doh }qrvfl rg zduwrvfl sdudphwux p 5 U/ rnuhvorqrvf irup|

nzdgudwrzhm i = U2 $ U/ mhvol=

d, i+{, @ +p2 4,{2. 5{22. 5p{{2/ e, i+{, @ +7  p2,{2. {22. 5p{{21

571 Vsurzdg}lf gr srvwdfl ndqrqlf}qhm irup|=

d, i = U $ U/ i+{, @ {{2. {{ . {2{ >

e, i = U $ U/ i+{, @ {2 {22. 5{2  7{{2. 7{{ >

f, i = Ue $ U/ i+{, @ 5{22. {2 . {2e 5{{e. 5{2{e1 4161 Lorf}|q vndoduq|

581 Qlhfk j = U  U $ U e hg}lh ixqnfmrqd­hp rnuhvorq|p z}ruhp j+{> |, @ 5{|. 6{2|2. { |  5{|2 5{2| =

Vsudzg}lf/ f}| j mhvw=

d, ixqnfmrqd­hp gzxolqlrz|p/

e, ixqnfmrqd­hp v|phwu|f}q|p/

f, lorf}|qhp vndoduq|p1

(2)

591 Vsudzg}lf/ f}| srgdqh rgz}rurzdqlh +m, mhvw lorf}|qhp vndoduq|p=

d, +m, = U  U $ U/ +{m|, @ {|. 5{2|2. { | >

e, +m, = U  U $ U/ +{m|, @ 6{|. 5{2|2. { | >

f, +m, = U  U $ U/ +{m|, @ 6{|. 5{2|2. { |  {|  { |>

g, +m, = F+kd> el,  F+kd> el, $ U/ +imj, @UK

@ i+w,j+w,gw1

5:1 Z|nd}df/ }h god nd }ghjr lorf}|qx vndoduqhjr +m, zsu}hvwu}hql U?lvwqlhmh wdnd pdflhu} T v|phwu|f}qd l grgdwqlr rnuhvorqd/ l } +{m|, @ {AT|1

5;1 Vsudzg}lf/ f}| ixqnfmrqd­ gzxolqlrz| j = U  U $ U mhvw lorf}|qhp vndoduq|p l z|}qdf}|f mhjr pdflhu} zed}lh xsru} dgnrzdqhm E su}hvwu}hql U / mhvol=

d, j+{> |, @ 5{|. 6{2|2. { | . 5{|2. 5{2| {2|  { |2/ E @

3 C 5

7 4

43 6 8 >

5 7 4

54 6 8 >

5 7 3

43 6 8

4 D>

e, j+{> |, @ {|. 5{2|2. 5{ |  {|2 {2|. 6{2| . 6{ |2/ E @

3 C 5 7 4

44 6 8 >

5 7 3

54 6 8 >

5

7 3

44 6 8

4 D>

f, j+{> |, @ {|. 5{2|2. 6{ |  {|2 {2|. {2| . { |2/ E @

3 C 5 7 4

44 6 8 >

5 7 3

54 6 8 >

5 7 3

44 6 8

4 D1

5<1 Ixqnfmrqd­ gzxolqlrz| j = U  U $ U pd zed}lh xsru} dgnrzdqhm 3

C 5

7 4

43 6 8 >

5 7 4

54 6 8 >

5 7 3

43 6 8

4

D pdflhu} D @ 5

7 5 4 4

4 6 3

4 3 4

6 8 = d, Vsudzg}lf/ f}| j mhvw lorf}|qhp vndoduq|p1

e, Z|}qdf}|f pdflhu} j zed}lh ndqrqlf}qhm su}hvwu}hql U 1

631 Qlhfk j = U  U $ U e hg}lh ixqnfmrqd­hp gzxolqlrz|p1 Z|nd}df/ }h j qlh mhvw ixqnfmrqd­hp v|phwu|f}q|p/ vsudzg}lf/ f}| ixqnfmrqd­

k+{> |, @ j+{> |, . j+|> {,

mhvw lorf}|qhp vndoduq|p/ z|}qdf}|f ed} h su}hvwu}hql U / znwöuhm k pd pdflhu}

gldjrqdoq d/ mhvol=

d, j+{> |, @ {| {2|2. 6{ |  7{2|. 7{ |2/ e, j+{> |, @ {|. 5{2|2 { |  {|2. 5{2| 1

(3)

641 Qlhfk j = U  U $ U e hg}lh ixqnfmrqd­hp gzxolqlrz|p1 Vsudzg}lf/

f}| ixqnfmrqd­ gzxolqlrz| k = U  U $ U rnuhvorq| z}ruhp k +{> |, @ 2+j +{> |, . j +|> {,,

mhvw lorf}|qhp vndoduq|p rud} z|}qdf}|f pdflhu} k zed}lh E su}hvwu}hql U / mhvol=

d, j+{> |, @ 5{|. 6{2|2. { | . 7{|2. 9{2| / E @

3 C 5

7 4

43 6 8 >

5 7 4

54 6 8 >

5 7 3

43 6 8

4 D>

e, j+{> |, @ {|. 7{2|2. 5{ |  5{|2. 5{2| / E @

3 C 5

7 4

43 6 8 >

5 7 3

43 6 8 >

5 7 4

54 6 8

4 D1

651 Qlhfk j = U  U $ U e hg}lh ixqnfmrqd­hp gzxolqlrz|p1 God md0 nlfk zduwrvfl sdudphwux p 5 U ixqnfmrqd­ j mhvw ixqnfmrqd­hp v|phwu|f}q|p/

lorf}|qhp vndoduq|p/ mhvol=

d, j+{> |, @ 5{|. 6{2|2. 5p{ |  p{|2 p{2|/ e, j+{> |, @ 5{|. 6{2|2. { |  5p{|  +p  5,{ |/

f, j+{> |, @ {|. 7{2|2. 5{ |  p{2|  +6  p,{ |2/

g, j+{> |, @ {|. 7{2|2. 5p | . {| . { |. p{2| . p{ |2B

661 Reolf}|f frvlqxv n dwd pl hg}| zhnwrudpl z su}hvwu}hql U } lorf}|qhp vndoduq|p +{m|, @ {A|/ mhvol=

d, { @ 5 7 4

55 6 8 > | @

5

7 3

45 6 8>

e, { @ 5

7 5

63 6 8 > | @

5

7 4

55 6 8>

f, { @ 5 7

s5 5s4

5 6 8 > | @

5 7 4

53 6 81

671 Reolf}|f frvlqxv n dwd pl hg}| zhnwrudpl d l e/ jg}lh=

d, d @ { . 5|/ e @ 6{  |/ mhvol zhnwru| { l | wzru} d ed} h ruwrqrupdoq d u}hf}|zlvwhm su}hvwu}hql xqlwduqhm Y >

e, d @ 5{  |/ e @ 5{ . |/ mhvol n{n @ n|n @ 5/ +{m|, @ 4>

f, d @ { . 5|/ e @ 5{  |/ mhvol n{n @ 4/ n|n @s

6 l n dw pl hg}| zhnwrudpl { l | mhvw uözq| S>

g, d @ 5{  |/ e @ { . 5|/ mhvol n{n @ n|n @ 4 l n dw pl hg}| zhnwrudpl { l | mhvw uözq| 2 >

(4)

h, d @ {  5|/ e @ 6{ . |/ mhvol n{n @ n|n @ 4 l n dw pl hg}| zhnwrudpl { l | mhvw uözq|  >

i, d @ 5{  6|/ e @ { . 7|/ mhvol n{n @ 5  n|n 9@ 3 l n dw pl hg}| zhnwrudpl { l | mhvw uözq|  >

j, d @ { . 6|/ e @ 5{  |/ mhvol n|n @ 5  n{n 9@ 3 l n dw pl hg}| zhnwrudpl { l | mhvw uözq| S>

k, d @ 5{ . |/ e @ { . |/ mhvol n|n @ 5  n{n 9@ 3 l n dw pl hg}| zhnwrudpl { l | mhvw uözq| 2 >

l, d @ 5{ . |/ e @ {  6|/ mhvol n{n @ 7  n|n 9@ 3 l n dw pl hg}| zhnwrudpl { l | mhvw uözq|  >

m, d @ {  |/ e @ {  5|/ mhvol n{n @ 7  n|n 9@ 3 l n dw pl hg}| zhnwrudpl { l | mhvw uözq|  1

681 Xgrzrgqlf/ }h mhvol qlh}hurzh zhnwru| {/ |/ } v d sdudpl ruwrjrqdoqh/ wr v d olqlrzr qlh}doh }qh1

691 Xgrzrgqlf/ }h mhvol qlh}hurzh zhnwru| {> {2> ===> {& v d sdudpl ruwrjrqdo0 qh/ wr v d olqlrzr qlh}doh }qh1

6:1 Reolf}|f qrup h zhnwrud 88 {/ jg}lh { 9@ 31

6;1 Z|nd}df/ }h mhvol E @ iy> y2> ===> y?j mhvw ed} d ruwrqrupdoq d su}hvwu}hql Y / wr god nd }ghjr { 5 Y =

d, { @ +{my, y. +{my2, y2. = = = . +{my?, y?/ e, n{n2 @ +{my,2. +{my2,2. = = = . +{my?,21

6<1 Xgrzrgqlf/ }h zgrzroqhm su}hvwu}hql xqlwduqhm Y =

d, +{m|, @ 3 / n{ . |n2 @ n{n2. n|n2/ srgdf lqwhusuhwdfm h jhrphwu|f}q d whjr wzlhug}hqld god Y @ U2>

e, mhvol +{m{, @ 3 god l 9@ m/ l> m @ 4> 5> ===> n/ wr

n{. {2. === . {&n2 @ n{n2. n{2n2. === . n{&n2>

f, n{ . |n2. n{  |n2 @ 5

n{n2. n|n2

+sudzr uözqrohj­rernx,1

731 Z|nd}df/ }h mhvol d 9@ e l n{  dn @ n{  en @ u/ wr{2+d . e, ? u1 741 Qlhfk { l | e hg d zhnwrudpl qlh}hurz|pl1 Xgrzrgqlf/ }h mhvol n{ . |n @

@ n{n . n|n/ wr lvwqlhmh  A 3 wdnlh/ }h | @ {1 F}| sudzg}lzh mhvw wzlhug}hqlh rgzurwqhB

751 Reolf}|f z duwrvf eh}z}jo hgq d frvlqxvd n dwd pl hg}| zhnwrudpl d @ 5{ . 6| 9@ 3 l e @ {  | 9@ 3>

mhvol zhnwru| { l | vsh­qldm d z duxqhn +{m|, @ n{n  n|n 9@ 31

(5)

761 Qlhfk Y e hg}lh su}hvwu}hql d xqlwduq d/ {> | 5 Y 1 Olf}e h g+{> |, @ n{  |n qd}|zdp| rgohj­rvfl d zhnwruöz { l |1 Xgrzrgqlf/ }h ixqnfmd g = Y  Y $ U rnuhvorqd z}ruhp g+{> |, @ n{  |n vsh­qld zduxqnl=

d, Y

cMTg+{> |,  3 a +g+{> |, @ 3 / { @ |, >

e, Y

cMTg+{> |, @ g+|> {,>

f, Y

cc!MTg+{> |,  g+{> }, . g+}> |,=

771 Reolf}|f rgohj­rvf pl hg}| zhnwrudpl #+w,/ #2+w, zsu}hvwu}hql zv}|vw0 nlfk zlhorpldqöz qdg fld­hp olf}e u}hf}|zlvw|fk } lorf}|qhp vndoduq|p rnuhvor0 q|p z}ruhp +*m*2, @U2

f *+w,*2+w,gw/ mhvol=

d, #+w, @ 5w2. 6w/ #2+w, @ w . 6/

e, #+w, @ w2 5w/ #2+w, @ w  61

781 Vsudzg}lf/ f}| srgdqh pdflhu}h v d ruwrjrqdoqh=

d,

% I2 I2

2 2

&

/ e,

 frv * vlq *

 vlq * frv *

 / f,

5

97 I2 3 4

4 3 3

3 I2 4 6 :81

791 Srgdf su}|n­dg wdnlhm pdflhu}| ruwrjrqdoqhm vwrsqld 6/ }h mhm slhuzv}d nroxpqd mhvw zhnwruhp d @

5 7

2

s5

2s3 5

6 81

7:1 Z|nd}df/ }h mhvol pdflhu}h D l E v d ruwrjrqdoqh/ wr pdflhu} DE wh } mhvw ruwrjrqdoqd1 F}| sudzg}lzh mhvw wzlhug}hqlh rgzurwqhB

7;1 Qlhfk D/ D2 e hg d pdflhu}dpl ruwrjrqdoq|pl1 F}| pdflhu}

D @

 D R R D2



mhvw ruwrjrqdoqdB

7<1 Qlhfk D l E e hg d pdflhu}dpl ruwrjrqdoq|pl vwrsqld 61 F}| z|qlnd vw dg/

}h pdflhu} F mhvw ruwrjrqdoqd/ mhvol=

d, F @ 2+D . E,/ e, F @ 5D  EB

831 Qlhfk D> E e hg d pdflhu}dpl ruwrjrqdoq|pl vwrsqld q A 41 F}| z|qlnd vw dg/ }h pdflhu} F mhvw ruwrjrqdoqd/ mhvol=

d, F @ DEA/ e, F @ DE/ f, F @ DE/ g, F @ DE/ h, F @ 6D  5E/ i, F @ 2D.2E/ j, F @ 2 D . EB

(6)

841 Qlhfk i = Ue $ Uee hg}lh su}hnv}wd­fhqlhp olqlrz|p rnuhvorq|p z}ruhp

i+{, @ 5 99 7

I22{. I22{

{e I2

2 {.I22{

{2

6 :: 8 =

Reolf}|f frvlqxv n dwd pl hg}| zhnwrudpl i+h,/ i+h.s

5h2,/ jg}lh h r}qdf}d m0w| +m @ 4> 5> 6> 7, zhnwru mhgqrvwnrz|1

851 God mdnlhm zduwrvfl sdudphwux d 5 U su}hnv}wd­fhqlh olqlrzh i = U $ U vsh­qld zduxqhn ni+{,n @ n{n/ mhvol=

d, i+{, @ 5

7 d{. d{

d{. d{

{2

6 8/

e, i +{, @ 5 7

2d{.2{2

{

2{ d{2

6 8/

f, i +{, @ 5 7

ed{.e{

e{. d{

{2

6 8B

861 Qlhfk i = U? $ U? e hg}lh su}hnv}wd­fhqlhp olqlrz|p vsh­qldm df|p zduxqhn ni +{,n @ n{n god nd }ghjr { 5 U?1 Reolf}|f frvlqxv n dwd pl hg}|

zhnwrudpl i+h,> i+5h. h2,/ jg}lh h> h2 v d rgsrzlhgqlr slhuzv}|p l guxjlp zhnwruhp mhgqrvwnrz|p z su}hvwu}hql U?1

871 God mdnlhm zduwrvfl sdudphwux p 5 U su}hnv}wd­fhqlh olqlrzh i = U $ U mhvw l}rphwul d olqlrz d/ mhvol=

d, i+{, @ 5

7 p{. p{

p{. p{

{2

6 8/

e, i+{, @ 5

7 +4  p,+{. { , {2

+p  4,{. +4  p,{

6 8B

881 God mdnlhm zduwrvfl sdudphwux p 5 U su}hnv}wd­fhqlh olqlrzh i = Ue$ Ue rnuhvorqh z}ruhp

i+{, @ 5 99 7

frv   {. vlq   {

p{2 p{e

 vlq   {. frv   {

p{2. p{e

6 :: 8

mhvw l}rphwul d olqlrz dB

(7)

891 Xgrzrgqlf/ }h mhvol su}hnv}wd­fhqlh i = U? $ U? mhvw l}rphwul d olqlrz d/

wr=

d, su}hnv}wd­fhqlh i mhvw qlhrvreolzh/

e, su}hnv}wd­fhqlh i mhvw l}rphwul d olqlrz d1

8:1 Qlhfk i = U? $ U?> j = U? $ U?/ jg}lh j mhvw elmhnfm d/ ehg d su}hnv}0 wd­fhqldpl olqlrz|pl1 F}| } zduxqnx/ }h j i mhvw l}rphwul d olqlrz d z |qlnd/

}h i mhvw l}rphwul d olqlrz dB Rgsrzlhg} x}dvdgqlf1

8;1 Qlhfk i = U? $ U?> j = U? $ U?/ jg}lh i mhvw elmhnfm d/ ehg d su}hnv}0 wd­fhqldpl olqlrz|pl1 F}| } zduxqnx/ }h j  i mhvw l}rphwul d olqlrz d z |qlnd/

}h j mhvw l}rphwul d olqlrz dB Rgsrzlhg} x}dvdgqlf1

8<1 Qlhfk Z e hg}lh srgsu}hvwu}hql d olqlrz d su}hvwu}hql Y 1 Z|nd}df/ }h Zz mhvw srgsu}hvwu}hql d olqlrz d su}hvwu}hql Y 1

931 Z|}qdf}|f }elöu zv}|vwnlfk zhnwruöz ruwrjrqdoq|fk gr srgsu}hvwu}hql olqlrzhm Z @ O+d,/ mhvol=

d, d @

4 5 4 A

>

e, d @

5 3 4 5 A 1

941 Z|}qdf}|f x}xsh­qlhqlh ruwrjrqdoqh srgsu}hvwu}hql Z/ mhvol=

d, Z @

{ 5 Ue = {. {2. {  {e @ 3 / e, Z @

{ 5 Ue = 5{. {2 { @ 3 a {. {2. {e @ 3 1 416141 Ruwrjrqdol}dfmd Judpd0Vfkplgwd

951 Z|}qdf}|f ed} h ruwrjrqdoq d su}hvwu}hql Z  U?} lorf}|qhp vndoduq|p rnuhvorq|p z}ruhp +{m|, @ {A|/ jg}lh=

d, Z @ O 3 C 5 7 3

34 6 8 >

5 7 4

33 6 8

4 D/

e, Z @ O 3 C 5 7 4

44 6 8 >

5

7 5

43 6 8

4 D/

f, Z @ O 3 C 5

7 3

45 6 8 >

5 7 3

44 6 8 >

5 7 4

43 6 8

4 D/

g, Z @ O 3 C 5

7 5

43 6 8 >

5 7 4

44 6 8 >

5

7 5

43 6 8

4 D/

h, Z @ O 3 C 5 7 4

53 6 8 >

5

7 4

44 6 8 >

5 7 4

44 6 8 >

5 7 5

44 6 8

4 D/

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