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2 Null Spaces, Column Spaces, and Linear Transformations

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4.2

Null Spaces, Column Spaces, and Linear Transformations

199

Inmatrixform, thissystemiswrittenas Ax D 0, where

Ä

1

5

3

9

2

1

(2)

Recallthatthesetofall x thatsatisfy(1)iscalledthe solutionset ofthesystem (1).

Oftenitisconvenienttorelatethissetdirectlytothematrix A andtheequation Ax D 0.

Wecallthesetof x thatsatisfy Ax D 0 the nullspace ofthematrix A.

DEFINITION

The nullspace ofan m n matrix A, writtenas Nul A, isthesetofallsolutions

ofthehomogeneousequation Ax D 0. Insetnotation,

Nul A D fx W x isin Rn and Ax D 0g

A moredynamicdescriptionof Nul A isthesetofall x in Rn thataremappedinto

thezerovectorof Rm viathelineartransformation x 7! Ax. SeeFig. 1.

lA

0

Nu

0

n

‫ޒ‬m

‫ޒ‬

FIGURE 1

2

3

5

EXAMPLE 1 Let A bethematrixin(2)above, andlet u D 4 3 5. Determineif

2

u belongstothenullspaceof A.

SOLUTION Totestif u satisﬁes Au D 0, simplycompute

Au D

Ä

1

5

3

9

2

3

Ä

5

2 4

5

9C4

35 D

25 C 27 2

1

2

D

Ä

0

0

Thus u isin Nul A.

Theterm space in nullspace isappropriatebecausethenullspaceofamatrixisa

vectorspace, asshowninthenexttheorem.

THEOREM 2

Thenullspaceofan m n matrix A isasubspaceof Rn . Equivalently, the

setofallsolutionstoasystem Ax D 0 of m homogeneouslinearequationsin

n unknownsisasubspaceof Rn .

200

CHAPTER 4

Vector Spaces

PROOF Certainly Nul A isasubsetof Rn because A has n columns. Wemustshow

that Nul A satisﬁesthethreepropertiesofasubspace. Ofcourse, 0 isin Nul A. Next,

let u and v representanytwovectorsin Nul A. Then

Au D 0

and

Av D 0

Toshowthat u C v isin Nul A, wemustshowthat A.u C v/ D 0. Usingapropertyof

matrixmultiplication, compute

A.u C v/ D Au C Av D 0 C 0 D 0

Thus u C v isin Nul A, and Nul A isclosedundervectoraddition. Finally, if c isany

scalar, then

A.c u/ D c.Au/ D c.0/ D 0

whichshowsthat c u isin Nul A. Thus Nul A isasubspaceof Rn .

EXAMPLE 2 Let H bethesetofallvectorsin R4 whosecoordinates a, b , c , d

satisfytheequations a

R4 .

2b C 5c D d and c

a D b . Showthat H isasubspaceof

SOLUTION Rearrangetheequationsthatdescribetheelementsof H , andnotethat H

isthesetofallsolutionsofthefollowingsystemofhomogeneouslinearequations:

a

a

2b C 5c

bC c

d D0

D0

ByTheorem2, H isasubspaceof R4 .

Itisimportantthatthelinearequationsdeﬁningtheset H arehomogeneous.

Otherwise, thesetofsolutionswilldeﬁnitely not beasubspace(becausethezerovector

isnotasolutionofanonhomogeneoussystem). Also, insomecases, thesetofsolutions

couldbeempty.

An Explicit Description of Nul A

Thereisnoobviousrelationbetweenvectorsin Nul A andtheentriesin A. Wesaythat

Nul A isdeﬁned implicitly, becauseitisdeﬁnedbyaconditionthatmustbechecked.

Noexplicitlistordescriptionoftheelementsin Nul A isgiven. However, solving

theequation Ax D 0 amountstoproducingan explicit descriptionof Nul A. Thenext

examplereviewstheprocedurefromSection1.5.

EXAMPLE 3 Findaspanningsetforthenullspaceofthematrix

2

3

2

6

2

4

1

2

5

1

3

8

3

7

15

4

SOLUTION Theﬁrststepistoﬁndthegeneralsolutionof Ax D 0 intermsoffree

variables. Rowreducetheaugmentedmatrix Œ A 0  to reduced echelonforminorder

towritethebasicvariablesintermsofthefreevariables:

2

3

x1 2x2

x4 C 3x5 D 0

1 2 0

1 3

0

40

0 1

2

2 0 5;

x3 C 2x4 2x5 D 0

0 0 0

0

0 0

0D0

4.2

Null Spaces, Column Spaces, and Linear Transformations

201

Thegeneralsolutionis x1 D 2x2 C x4 3x5 , x3 D 2x4 C 2x5 , with x2 , x4 , and x5

free. Next, decomposethevectorgivingthegeneralsolutionintoalinearcombination

ofvectorswhere theweightsarethefreevariables. Thatis,

2 3 2

3

2

3

2

3

2 3

x1

2x2 C x4 3x5

1

3

2

6 x2 7 6

6 07

6 07

7

617

x2

6 7 6

6

6

7

7

7

6 7

6 x3 7 D 6

7

6

7

6

7

6

7

2x

C

2x

2

0

C

x

C

x

D

x

4

5 7

46

56 2 7

26 7

6 7 6

7

4 x4 5 4

5

405

4 15

4 05

x4

x5

x5

0

1

0

D x2 u C x4 v C x5 w

"

u

"

v

"

w

(3)

Everylinearcombinationof u, v, and w isanelementof Nul A. Thus fu; v; wg isa

spanningsetfor Nul A.

problemsofthistypewhere Nul A containsnonzerovectors. Wewillusethesefacts

later.

1. ThespanningsetproducedbythemethodinExample 3isautomaticallylinearly

independentbecausethefreevariablesaretheweightsonthespanningvectors. For

instance, lookatthe2nd, 4th, and5thentriesinthesolutionvectorin(3)andnote

that x2 u C x4 v C x5 w canbe 0 onlyiftheweights x2 ; x4 , and x5 areallzero.

2. When Nul A containsnonzerovectors, thenumberofvectorsinthespanningsetfor

Nul A equalsthenumberoffreevariablesintheequation Ax D 0.

The Column Space of a Matrix

Anotherimportantsubspaceassociatedwithamatrixisitscolumnspace. Unlikethe

nullspace, thecolumnspaceisdeﬁnedexplicitlyvialinearcombinations.

DEFINITION

The columnspace ofan m n matrix A, writtenas Col A, isthesetofalllinear

an , then

combinationsofthecolumnsof A. If A D Œ a1

Col A D Span fa1 ; : : : ; an g

Since Span fa1 ; : : : ; an g isasubspace, byTheorem 1, thenexttheoremfollowsfrom

thedeﬁnitionof Col A andthefactthatthecolumnsof A arein Rm .

THEOREM 3

Thecolumnspaceofan m

n matrix A isasubspaceof Rm .

Notethatatypicalvectorin Col A canbewrittenas Ax forsome x becausethe

notation Ax standsforalinearcombinationofthecolumnsof A. Thatis,

Col A D fb W b D Ax forsome x in Rn g

Thenotation Ax forvectorsin Col A alsoshowsthat Col A isthe range ofthelinear

transformation x 7! Ax. Wewillreturntothispointofviewattheendofthesection.

202

CHAPTER 4

Vector Spaces

EXAMPLE 4 Findamatrix A suchthat W D Col A.

x2

82

9

3

< 6a b

=

W D 4 a C b 5 W a, b in R

:

;

7a

x3

0

W

x1

SOLUTION First, write W asasetoflinearcombinations.

8 2

9

82

3

2

3

3 2

39

6

1

6

1 =

<

=

<

W D a4 1 5 C b 4 1 5 W a, b in R D Span 4 1 5; 4 1 5

:

;

:

;

7

0

7

0

2

6

Second, usethevectorsinthespanningsetasthecolumnsof A. Let A D 4 1

7

Then W D Col A, asdesired.

3

1

1 5.

0

RecallfromTheorem 4inSection1.4thatthecolumnsof A span Rm ifandonlyif

theequation Ax D b hasasolutionforeach b. Wecanrestatethisfactasfollows:

Thecolumnspaceofan m n matrix A isallof Rm ifandonlyiftheequation

Ax D b hasasolutionforeach b in Rm .

The Contrast Between Nul A and Col A

Itisnaturaltowonderhowthenullspaceandcolumnspaceofamatrixarerelated. In

fact, thetwospacesarequitedissimilar, asExamples 5–7willshow. Nevertheless,

a surprising connection between the null space and column space will emerge in

Section 4.6, aftermoretheoryisavailable.

EXAMPLE 5 Let

2

2

3

4

5

7

2

7

8

3

1

35

6

a. Ifthecolumnspaceof A isasubspaceof Rk , whatis k ?

b. Ifthenullspaceof A isasubspaceof Rk , whatis k ?

SOLUTION

a. Thecolumnsof A eachhavethreeentries, so Col A isasubspaceof Rk , where

k D 3.

b. A vector x suchthat Ax isdeﬁnedmusthavefourentries, so Nul A isasubspaceof

Rk , where k D 4.

Whenamatrixisnotsquare, asinExample 5, thevectorsin Nul A and Col A live

inentirelydifferent“universes.” Forexample, nolinearcombinationofvectorsin R3

canproduceavectorin R4 . When A issquare, Nul A and Col A dohavethezerovector

incommon, andinspecialcasesitispossiblethatsomenonzerovectorsbelongtoboth

Nul A and Col A.

4.2

Null Spaces, Column Spaces, and Linear Transformations

203

EXAMPLE 6 With A asinExample 5, ﬁndanonzerovectorin Col A andanonzero

vectorin Nul A.

2

3

2

SOLUTION Itiseasytoﬁndavectorin Col A. Anycolumnof A willdo, say, 4 2 5.

3

Toﬁndanonzerovectorin Nul A, rowreducetheaugmentedmatrix Œ A 0  andobtain

ŒA 0

2

1

40

0

0

1

0

9

5

0

0

0

1

3

0

05

0

Thus, if x satisﬁes Ax D 0, then x1 D 9x3 , x2 D 5x3 , x4 D 0, and x3 isfree. Assigninganonzerovalueto x3 —say, x3 D 1—weobtainavectorin Nul A, namely,

x D . 9; 5; 1; 0/.

2

3

2

3

3

3

6 27

7

4 1 5.

EXAMPLE 7 With A asinExample 5, let u D 6

4 1 5 and v D

3

0

a. Determineif u isin Nul A. Could u bein Col A?

b. Determineif v isin Col A. Could v bein Nul A?

SOLUTION

a. Anexplicitdescriptionof Nul A

Au.

2

2 4

Au D 4 2 5

3 7

isnotneededhere. Simplycomputetheproduct

2

7

8

2

3

3

2

3 2 3

3

1 6

0

0

7

27 4

5 Ô 405

3 56

3

D

4 15

6

3

0

0

Obviously, u is not asolutionof Ax D 0, so u isnotin Nul A. Also, withfourentries,

u couldnotpossiblybein Col A, since Col A isasubspaceof R3 .

b. Reduce Œ A v  toanechelonform.

2

3 2

3

2 4

2 1 3

2 4

2 1 3

5 4 25

ŒA v D 4 2 5 7 3 15 40 1

3 7

8 6 3

0 0 0 17 1

Atthispoint, itisclearthattheequation Ax D v isconsistent, so v isin Col A. With

onlythreeentries, v couldnotpossiblybein Nul A, since Nul A isasubspaceof

R4 .

Thetableonpage204summarizeswhatwehavelearnedabout Nul A and Col A.

Item 8isarestatementofTheorems 11and12(a)inSection 1.9.

Kernel and Range of a Linear Transformation

Subspacesofvectorspacesotherthan Rn areoftendescribedintermsofalinear

giveninSection 1.8.

204

CHAPTER 4

Vector Spaces

Contrast Between Nul A and Col A for an m x n Matrix A

Nul A

Col A

1. Nul A isasubspaceof R .

1. Col A isasubspaceof Rm .

2. Nul A isimplicitlydeﬁned; thatis, youare

givenonlyacondition .Ax D 0/ thatvectorsin Nul A mustsatisfy.

2. Col A isexplicitlydeﬁned; thatis, youare

toldhowtobuildvectorsin Col A.

3. Ittakestimetoﬁndvectorsin Nul A. Row

operationson Œ A 0  arerequired.

3. Itiseasytoﬁndvectorsin Col A. The

columns of A are displayed; others are

formedfromthem.

4. Thereisnoobviousrelationbetween Nul A

andtheentriesin A.

4. Thereisanobviousrelationbetween Col A

andtheentriesin A, sinceeachcolumnof

A isin Col A.

5. A typicalvector v in Nul A hastheproperty

that Av D 0.

5. A typicalvector v in Col A hastheproperty

thattheequation Ax D v isconsistent.

n

6. Givenaspeciﬁcvector v, itiseasytotellif

v isin Nul A. Justcompute Av.

6. Givenaspeciﬁcvector v, itmaytaketime

totellif v isin Col A. Rowoperationson

Œ A v  arerequired.

7. Nul A D f0g ifandonlyiftheequation

Ax D 0 hasonlythetrivialsolution.

7. Col A D Rm ifandonlyiftheequation

Ax D b hasasolutionforevery b in Rm .

8. Nul A D f0g ifandonlyifthelineartransformation x 7! Ax isone-to-one.

DEFINITION

8. Col A D Rm ifandonlyifthelineartransformation x 7! Ax maps Rn onto Rm .

A lineartransformation T fromavectorspace V intoavectorspace W isarule

thatassignstoeachvector x in V auniquevector T .x/ in W , suchthat

(i) T .u C v/ D T .u/ C T .v/

(ii) T .c u/ D cT .u/

forall u, v in V , and

forall u in V andallscalars c .

The kernel (or nullspace)ofsucha T isthesetofall u in V suchthat T .u/ D 0

(thezerovectorin W /. The range of T isthesetofallvectorsin W oftheform T .x/

forsome x in V . If T happenstoariseasamatrixtransformation—say, T .x/ D Ax

forsomematrix A—thenthekernelandtherangeof T arejustthenullspaceandthe

columnspaceof A, asdeﬁnedearlier.

Itisnotdifﬁculttoshowthatthekernelof T isasubspaceof V . Theproofis

essentiallythesameastheoneforTheorem 2. Also, therangeof T isasubspaceof W .

SeeFig. 2andExercise 30.

in

a

om

D

T

l

ne

r

Ke

0

Ra

ng

e

0

W

V

Kernel is a

subspace of V

Range is a

subspace of W

FIGURE 2 Subspacesassociatedwith

alineartransformation.

Inapplications, asubspaceusuallyarisesaseitherthekernelortherangeofan

appropriatelineartransformation. Forinstance, thesetofallsolutionsofahomogeneouslineardifferentialequationturnsouttobethekernelofalineartransformation.

4.2

Null Spaces, Column Spaces, and Linear Transformations

205

Typically, suchalineartransformationisdescribedintermsofoneormorederivatives

ofafunction. Toexplainthisinanydetailwouldtakeustoofaraﬁeldatthispoint. So

weconsideronlytwoexamples. Theﬁrstexplainswhytheoperationofdifferentiation

isalineartransformation.

EXAMPLE 8 (Calculus required) Let V bethevectorspaceofallreal-valuedfunc-

tions f deﬁnedonaninterval Œa; b withthepropertythattheyaredifferentiableand

theirderivativesarecontinuousfunctionson Œa; b. Let W bethevectorspace C Œa; b

ofallcontinuousfunctionson Œa; b, andlet D W V ! W bethetransformationthat

changes f in V intoitsderivative f 0 . Incalculus, twosimpledifferentiationrulesare

D.f C g/ D D.f / C D.g/

and

D.cf / D cD.f /

Thatis, D isalineartransformation. Itcanbeshownthatthekernelof D isthesetof

constantfunctionson Œa; b andtherangeof D istheset W ofallcontinuousfunctions

on Œa; b.

EXAMPLE 9 (Calculus required) Thedifferentialequation

y 00 C ! 2 y D 0

(4)

where ! isaconstant, isusedtodescribeavarietyofphysicalsystems, suchasthe

vibrationofaweightedspring, themovementofapendulum, andthevoltageinan

inductance-capacitanceelectricalcircuit. Thesetofsolutionsof(4)ispreciselythe

kernelofthelineartransformationthatmapsafunction y D f .t / intothefunction

f 00 .t / C ! 2 f .t /. Findinganexplicitdescriptionofthisvectorspaceisaproblemin

differentialequations. ThesolutionsetturnsouttobethespacedescribedinExercise 19

inSection 4.1.

PRACTICE PROBLEMS

82 3

9

< a

=

1. Let W D 4 b 5 W a 3b c D 0 . Showintwodifferentwaysthat W isa

:

;

c

subspaceof R3 . (Usetwotheorems.)

2

3

2

3

2

3

7 3 5

2

7

5 5, v D 4 1 5, and w D 4 6 5. Supposeyouknowthat

2. Let A D 4 4 1

5 2

4

1

3

theequations Ax D v and Ax D w arebothconsistent. Whatcanyousayaboutthe

equation Ax D v C w?

4.2 EXERCISES

2

3

1

1. Determineif w D 4 3 5 isin Nul A, where

4

2

3

3

5

3

2

0 5:

8

4

1

2

3

1

2. Determineif w D 4 1 5 isin Nul A, where

1

2

3

2

6

4

2

5 5:

5

4

1

206

CHAPTER 4

Vector Spaces

InExercises3–6, ﬁndanexplicitdescriptionof Nul A, bylisting

vectorsthatspanthenullspace.

Ä

1

2

4

0

3. A D

0

1

3

2

Ä

1

3

2

0

4. A D

0

0

3

0

2

3

1

4

0

2

0

0

1

5

05

5. A D 4 0

0

0

0

0

2

2

3

1

3

4

3

1

1

3

1

05

6. A D 4 0

0

0

0

0

0

21. With A asinExercise 17, ﬁndanonzerovectorin Nul A and

anonzerovectorin Col A.

InExercises7–14, eitheruseanappropriatetheoremtoshowthat

thegivenset, W , isavectorspace, orﬁndaspeciﬁcexampleto

thecontrary.

9

82 3

9

82 3

< a

=

< r

=

8. 4 s 5 W 3r 2 D 3s C t

7. 4 b 5 W a C b C c D 2

:

;

:

;

c

t

9

82 3

9

82 3

a

p

>

ˆ

>

ˆ

>

ˆ

>

ˆ

=

<6 7

=

<6 7

b 7 3a C b D c

q 7 p 3q D 4s

6

6

10. 4 5 W

9. 4 5 W

c

a C b C 2c D 2d >

r

2p D s C 5r >

ˆ

ˆ

>

ˆ

>

ˆ

;

:

;

:

d

s

9

8

9

82

3

2

3

3p 5q

s 2t

>

ˆ

>

ˆ

>

ˆ

>

ˆ

=

<

=

<6

6 4q 7

3 C 3s 7

7 W p , q real

6

7 W s , t real

12.

11. 6

5

4

5

4

p

3s C t

>

ˆ

>

ˆ

>

ˆ

>

ˆ

;

:

;

:

qC1

2s

9

9

82

82

3

3

s C 3t

=

=

<

< c 6d

13. 4 d 5 W c , d real 14. 4 s 2t 5 W s , t real

;

;

:

:

5s t

c

InExercises25and26, A denotesan m n matrix. Markeach

InExercises15and16, ﬁnd A suchthatthegivensetis Col A.

82

9

3

2s C t

ˆ

>

ˆ

>

<6

=

r s C 2t 7

7

6

W

r

,

s

,

t

real

15. 4

3r C s 5

ˆ

>

ˆ

>

:

;

2r s t

9

82

3

b c

ˆ

>

>

ˆ

=

<6

7

2b

C

3d

7 W b , c , d real

16. 6

4

5

b C 3c 3d

ˆ

>

>

ˆ

;

:

cCd

ForthematricesinExercises17–20, (a)ﬁnd k suchthat Nul A is

asubspaceof Rk , and(b)ﬁnd k suchthat Col A isasubspaceof

Rk .

2

3

2

3

6

4

5

2

3

6 3

6 1

27

0

17

7

7

17. A D 6

18. A D 6

4 9

4 0

65

2

25

9

6

5

7

2

Ä

4

5

2

6

0

19. A D

1

1

0

1

0

20. A D 1

3

2

0

5

22. With A asinExercise18, ﬁndanonzerovectorin Nul A and

anonzerovectorin Col A.

Ä

Ä

2

4

2

23. Let A D

and w D

. Determineif w isin

1

2

1

Col A. Is w in Nul A?

3

2

2 3

10

8

2

2

2

6 0

7

627

2

2

2

7 and w D 6 7. Determine

24. Let A D 6

4 1

405

1

6

05

1

1

0

2

2

if w isin Col A. Is w in Nul A?

25. a. Thenullspaceof A isthesolutionsetoftheequation

Ax D 0.

b. Thenullspaceofan m

n matrixisin Rm .

c. The column space of A is the range of the mapping

x 7! Ax.

d. Iftheequation Ax D b isconsistent, then Col A is Rm .

e. Thekernelofalineartransformationisavectorspace.

f. Col A isthesetofallvectorsthatcanbewrittenas Ax for

some x.

26. a. A nullspaceisavectorspace.

b. Thecolumnspaceofan m

n matrixisin Rm .

c. Col A isthesetofallsolutionsof Ax D b.

d. Nul A isthekernelofthemapping x 7! Ax.

e. Therangeofalineartransformationisavectorspace.

f. Thesetofallsolutionsofahomogeneouslineardifferentialequationisthekernelofalineartransformation.

27. Itcanbeshownthatasolutionofthesystembelowis x1 D 3,

x2 D 2, and x3 D 1. Usethisfactandthetheoryfromthis

sectiontoexplainwhyanothersolutionis x1 D 30, x2 D 20,

and x3 D 10. (Observehowthesolutionsarerelated, but

makenoothercalculations.)

x1

3x2

3x3 D 0

2x1 C 4x2 C 2x3 D 0

x1 C 5x2 C 7x3 D 0

28. Considerthefollowingtwosystemsofequations:

5x1 C x2

3x3 D 0

5x1 C x2

3x3 D 0

4x1 C x2

6x3 D 9

4x1 C x2

6x3 D 45

9x1 C 2x2 C 5x3 D 1

9x1 C 2x2 C 5x3 D 5

Itcanbeshownthattheﬁrstsystemhasasolution. Use

thisfactandthetheoryfromthissectiontoexplainwhythe

secondsystemmustalsohaveasolution. (Makenorow

operations.)

4.2

Null Spaces, Column Spaces, and Linear Transformations

29. ProveTheorem3asfollows: Givenan m n matrix A, an

elementin Col A hastheform Ax forsome x in Rn . Let Ax

and Aw representanytwovectorsin Col A.

a. Explainwhythezerovectorisin Col A.

b. Showthatthevector Ax C Aw isin Col A.

c. Givenascalar c , showthat c.Ax/ isin Col A.

30. Let T W V ! W bealineartransformationfromavector

space V intoavectorspace W . Provethattherangeof T

isasubspaceof W . [Hint: Typicalelementsoftherange

havetheform T .x/ and T .w/ forsome x, w in V .]

Ä

p.0/

31. Deﬁne T W P2 ! R2 by T .p/ D

. Forinstance, if

p.1/

Ä

3

p.t/ D 3 C 5t C 7t 2 , then T .p/ D

.

15

a. Show that T is a linear transformation. [Hint: For

arbitrarypolynomials p, q in P2 , compute T .p C q/ and

T .c p/.]

b. Findapolynomial p in P2 thatspansthekernelof T , and

describetherangeof T .

32. Deﬁne Äa linear transformation T W P2 ! R2 by

p.0/

T .p / D

. Findpolynomials p1 and p2 in P2 that

p.0/

spanthekernelof T , anddescribethe rangeof T .

33. Let M2 2 be the vector space of all 2 2 matrices,

anddeﬁne

T W M2 2 ! M2 2 by T .A/ D A C AT , where

Ä

a

b

.

c

d

a. Showthat T isalineartransformation.

b. Let B beanyelementof M2 2 suchthat B T D B . Find

an A in M2 2 suchthat T .A/ D B .

c. Showthattherangeof T isthesetof B in M2

propertythat B T D B .

2

withthe

d. Describethekernelof T .

34. (Calculusrequired)Deﬁne T W C Œ0; 1 ! C Œ0; 1 asfollows:

For f in C Œ0; 1, let T .f/ betheantiderivative F of f such

that F.0/ D 0. Showthat T isalineartransformation, and

describethekernelof T . (SeethenotationinExercise20of

Section4.1.)

207

35. Let V and W bevectorspaces, andlet T W V ! W bealinear

transformation. Givenasubspace U of V , let T .U / denote

thesetofallimagesoftheform T .x/, where x isin U . Show

that T .U / isasubspaceof W .

36. Given T W V ! W asinExercise35, andgivenasubspace

Z of W , let U bethesetofall x in V suchthat T .x/ isin Z .

Showthat U isasubspaceof V .

37. [M] Determinewhether w isinthecolumnspaceof A, the

nullspaceof A, orboth, where

2

3

2

3

1

7

6

4

1

6 17

6 5

1

0

27

7

6

7

wD6

4 1 5; A D 4 9

11

7

35

3

19

9

7

1

38. [M] Determinewhether w isinthecolumnspaceof A, the

nullspaceof A, orboth, where

2 3

2

3

1

8

5

2

0

627

6 5

2

1

27

7

6

7

wD6

4 1 5; A D 4 10

8

6

35

0

3

2

1

0

39. [M] Let a1 ; : : : ; a5 denote the columns of the matrix A,

where

3

2

5

1

2

2

0

63

3

2

1

12 7

7; B D Œ a1 a2 a4 

48

4

4

5

12 5

2

1

1

0

2

a. Explainwhy a3 and a5 areinthecolumnspaceof B .

b. Findasetofvectorsthatspans Nul A.

c. Let T W R5 ! R4 bedeﬁnedby T .x/ D Ax. Explainwhy

T isneitherone-to-onenoronto.

40. [M] Let H D Span fv1 ; v2 g and K D Span fv3 ; v4 g, where

3

3

2

2

2 3

2 3

0

2

1

5

v1 D 4 3 5; v2 D 4 3 5; v3 D 4 1 5; v4 D 4 12 5:

28

5

4

8

Then H and K are subspaces of R3 . In fact, H and

K areplanesin R3 throughtheorigin, andtheyintersect

in a line through 0. Find a nonzero vector w that generatesthatline. [Hint: w canbewrittenas c1 v1 C c2 v2

andalsoas c3 v3 C c4 v4 . Tobuild w, solvetheequation

c1 v1 C c2 v2 D c3 v3 C c4 v4 fortheunknown cj ’s.]

SG

Mastering: Vector Space, Subspace,

Col A, and Nul A 4–6

SOLUTIONS TO PRACTICE PROBLEMS

1. Firstmethod: W isasubspaceof R3 byTheorem 2because W isthesetofallsolutionstoasystemofhomogeneouslinearequations(wherethesystemhasonlyone

3

1 .

equation). Equivalently, W isthenullspaceofthe 1 3 matrix A D Œ 1

208

CHAPTER 4

Vector Spaces

Secondmethod: Solvetheequation a

a in

2

3

3b C c

termsofthefreevariables b and c . Anysolutionhastheform 4 b 5, where b

c

and c arearbitrary, and

2

3

2 3

2 3

3b C c

1

3

4 b 5 D b4 1 5 C c4 0 5

c

0

1

3b

"

v1

"

v2

Thiscalculationshowsthat W D Span fv1 ; v2 g. Thus W isasubspaceof R3 by

Theorem1. Wecouldalsosolvetheequation a 3b c D 0 for b or c andget

alternativedescriptionsof W asasetoflinearcombinationsoftwovectors.

2. Both v and w arein Col A. Since Col A isavectorspace, v C w mustbein Col A.

Thatis, theequation Ax D v C w isconsistent.

4.3 LINEARLY INDEPENDENT SETS; BASES

Inthissectionweidentifyandstudythesubsetsthatspanavectorspace V orasubspace

H as“efﬁciently”aspossible. Thekeyideaisthatoflinearindependence, deﬁnedas

in Rn .

Anindexedsetofvectors fv1 ; : : : ; vp g in V issaidtobe linearlyindependent if

thevectorequation

c1 v1 C c2 v2 C C cp vp D 0

(1)

has only thetrivialsolution, c1 D 0; : : : ; cp D 0.¹

Theset fv1 ; : : : ; vp g issaidtobe linearlydependent if(1)hasanontrivialsolution,

thatis, iftherearesomeweights, c1 ; : : : ; cp , notallzero, suchthat(1)holds. Insucha

case, (1)iscalleda lineardependencerelation among v1 ; : : : ; vp .

Justasin Rn , asetcontainingasinglevector v islinearlyindependentifandonlyif

v Ô 0. Also, asetoftwovectorsislinearlydependentifandonlyifoneofthevectors

isamultipleoftheother. Andanysetcontainingthezerovectorislinearlydependent.

ThefollowingtheoremhasthesameproofasTheorem 7inSection 1.7.

THEOREM 4

Anindexedset fv1 ; : : : ; vp g oftwoormorevectors, with v1 Ô 0, islinearly

dependentifandonlyifsome vj (with j > 1/ isalinearcombinationofthe

precedingvectors, v1 ; : : : ; vj 1 .

Themaindifferencebetweenlineardependencein Rn andinageneralvectorspace

isthatwhenthevectorsarenot n-tuples, thehomogeneousequation(1)usuallycannot

thecolumnsofamatrix A inordertostudytheequation Ax D 0. Wemustrelyinstead

onthedeﬁnitionoflineardependenceandonTheorem 4.

EXAMPLE 1 Let p1 .t / D 1, p2 .t / D t , and p3 .t/ D 4

linearlydependentin P because p3 D 4p1

p2 .

t . Then fp1 ; p2 ; p3 g is

¹ Itisconvenienttouse c1 ; : : : ; cp in(1)forthescalarsinsteadof x1 ; : : : ; xp , aswedidinChapter 1.

4.3

Linearly Independent Sets; Bases

209

EXAMPLE 2 Theset fsin t; cos tg islinearlyindependentin C Œ0; 1, thespaceof

allcontinuousfunctionson 0 Ä t Ä 1, because sin t and cos t arenotmultiplesofone

another asvectorsin C Œ0; 1. Thatis, thereisnoscalar c suchthat cos t D c sin t for

all t in Œ0; 1. (Lookatthegraphsof sin t and cos t .) However, fsin t cos t; sin 2tg is

linearlydependentbecauseoftheidentity: sin 2t D 2 sin t cos t , forall t .

Let H be a subspace of a vector space V .

B D fb1 ; : : : ; bp g in V isa basis for H if

DEFINITION

An indexed set of vectors

(i) B isalinearlyindependentset, and

(ii) thesubspacespannedby B coincideswith H ; thatis,

H D Span fb1 ; : : : ; bp g

Thedeﬁnitionofabasisappliestothecasewhen H D V , becauseanyvectorspace

isasubspaceofitself. Thusabasisof V isalinearlyindependentsetthatspans V .

Observethatwhen H Ô V , condition(ii)includestherequirementthateachofthe

vectors b1 ; : : : ; bp mustbelongto H , because Span fb1 ; : : : ; bp g contains b1 ; : : : ; bp ,

asshowninSection 4.1.

an . Then

EXAMPLE 3 Let A beaninvertible n n matrix—say, A D Œ a1

thecolumnsof A formabasisfor Rn becausetheyarelinearlyindependentandthey

span Rn , bytheInvertibleMatrixTheorem.

EXAMPLE 4 Let e1 ; : : : ; en bethecolumnsofthe n

x3

is,

e3

2 3

1

607

6 7

e1 D 6 : 7;

4 :: 5

0

e2

e1

x1

FIGURE 1

Thestandardbasisfor R3 .

x2

2 3

0

617

6 7

e 2 D 6 : 7;

4 :: 5

0

:::;

n identitymatrix, In . That

2 3

0

6 :: 7

6 7

en D 6 : 7

405

1

Theset fe1 ; : : : ; en g iscalledthe standardbasis for Rn (Fig. 1).

2

3

2

3

2

3

3

4

2

EXAMPLE 5 Let v1 D 4 0 5, v2 D 4 1 5, and v3 D 4 1 5. Determineif

6

7

5

fv1 ; v2 ; v3 g isabasisfor R3 .

SOLUTION Sincethereareexactlythreevectorsherein R3 , wecanuseanyofseveral

methodstodetermineifthematrix A D Œ v1 v2 v3  isinvertible. Forinstance, two

rowreplacementsrevealthat A hasthreepivotpositions. Thus A isinvertible. Asin

Example3, thecolumnsof A formabasisfor R3 .

EXAMPLE 6 Let S D f1; t; t 2 ; : : : ; t n g. Verifythat S isabasisfor Pn . Thisbasis

iscalledthe standardbasis for Pn .

SOLUTION Certainly S spans Pn . Toshowthat S islinearlyindependent, supposethat

c0 ; : : : ; cn satisfy

c0 1 C c1 t C c2 t 2 C C cn t n D 0.t /

(2)

Thisequalitymeansthatthepolynomialonthelefthasthesamevaluesasthezeropolynomialontheright. A fundamentaltheoreminalgebrasaysthattheonlypolynomial

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