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Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeLinear Transformations

Math 240 | Calculus III

Summer 2013, Session II

Tuesday, July 23, 2013

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeAgenda

1.

Linea rT ransformations

Linear transformations of Euclidean space

2.

Ke rneland Range

3.

The matrix of a linea rtransfo rmation

Composition of linear transformations

Kernel and Range

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeMotivation

In themnlinear system

Ax=0; we can regardAas transforming elements ofRn(as column vectors) into elements ofRmvia the rule T(x) =Ax:Then solving the system amounts to nding all of the vectors x2Rnsuch thatT(x) =0.Solving the dierential equation y

00+y= 0

is equivalent to nding functionsysuch thatT(y) = 0, where

Tis dened as

T(y) =y00+y:

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeDenition

Denition

LetVandWbe vector spaces with the same scalars. A

mappingT:V!Wis called alinear transformationfrom

VtoWif it satises

1.T(u+v) =T(u) +T(v)and

2.T(cv) =cT(v)

for all vectorsu;v2Vand all scalarsc.Vis called the domainandWthecodomainofT.Examples I

T:Rn!Rmdened byT(x) =Ax, whereAis an

mnmatrixI

T:Ck(I)!Ck2(I)dened byT(y) =y00+yI

T:Mmn(R)!Mnm(R)dened byT(A) =ATI

T:P1!P2dened byT(a+bx) = (a+2b)+3ax+4bx2

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeExamples

1.

V erifythat T:Mmn(R)!Mnm(R), where

T(A) =AT, is a linear transformation.I

The transpose of anmnmatrix is annmmatrix.I

IfA;B2Mmn(R), then

T(A+B) = (A+B)T=AT+BT=T(A) +T(B):I

IfA2Mmn(R)andc2R, then

T(cA) = (cA)T=cAT=cT(A):2.V erifythat T:Ck(I)!Ck2(I), whereT(y) =y00+y, is a linear transformation.I

Ify2Ck(I)thenT(y) =y00+y2Ck2(I).I

Ify1;y22Ck(I), then

T(y1+y2) = (y1+y2)00+ (y1+y2) =y001+y002+y1+y2

= (y001+y1) + (y002+y2) =T(y1) +T(y2):I

Ify2Ck(I)andc2R, then

T(cy) = (cy)00+ (cy) =cy00+cy=c(y00+y) =cT(y):

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeSpecifying linear transformations

A consequence of the properties of a linear transformation is that they preserve linear combinations, in the sense that T(c1v1++cnvn) =c1T(v1) ++cnT(vn):In particular, iffv1;:::;vngis a basis for the domain ofT, then knowingT(v1);:::;T(vn)is enough to determineT everywhere.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeLinear transformations fromRntoRmLetAbe anmnmatrix with real entries and dene

T:Rn!RmbyT(x) =Ax. Verify thatTis a linear

transformation.I

Ifxis ann1column vector thenAxis anm1

column vector.I

T(x+y) =A(x+y) =Ax+Ay=T(x) +T(y)I

T(cx) =A(cx) =cAx=cT(x)Such a transformation is called amatrix transformation.In fact,everylinear transformation fromRntoRmis a matrix transformation.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeMatrix transformations

Theorem

LetT:Rn!Rmbe a linear transformation. ThenTis

described by the matrix transformationT(x) =Ax, where

A=T(e1)T(e2)T(en)

ande1;e2;:::;endenote the standard basis vectors forRn.ThisAis called thematrix ofT.Example Determine the matrix of the linear transformationT:R4!R3 dened by

T(x1;x2;x3;x4) = (2x1+ 3x2+x4;5x1+ 9x3x4;

4x1+ 2x2x3+ 7x4):

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeKernel

Denition

SupposeT:V!Wis a linear transformation. The set

consisting of all the vectorsv2Vsuch thatT(v) =0is called thekernelofT. It is denoted

Ker(T) =fv2V:T(v) =0g:Example

LetT:Ck(I)!Ck2(I)be the linear transformation

T(y) =y00+y. Its kernel is spanned byfcosx;sinxg.Remarks I The kernel of a linear transformation is a subspace of its domain.I The kernel of a matrix transformation is simply the null space of the matrix.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeRange

Denition

Therangeof the linear transformationT:V!Wis the

subset ofWconsisting of everything \hit by"T. In symbols,

Rng(T) =fT(v)2W:v2Vg:Example

Consider the linear transformationT:Mn(R)!Mn(R)

dened byT(A) =A+AT.The range ofTis the subspace of symmetricnnmatrices.Remarks I The range of a linear transformation is a subspace of its codomain.I The range of a matrix transformation is the column space of the matrix.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeRank-Nullity revisited

SupposeTis the matrix transformation withmnmatrixA.We know I

Ker(T) = nullspace(A),I

Rng(T) = colspace(A),I

the domain ofTisRn.Hence, I dim(Ker(T)) = nullity(A),I dim(Rng(T)) = rank(A),I dim(domain ofT) =n.We know from the rank-nullity theorem that rank(A) + nullity(A) =n:This fact is also true whenTis not a matrix transformation:Theorem

IfT:V!Wis a linear transformation andVis

nite-dimensional, then dim(Ker(T)) + dim(Rng(T)) = dim(V):

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeThe function of bases

Theorem

LetVbe a vector space with basisfv1;v2;:::;vng. Then every vectorv2Vcan be written in a unique wayas a linear combination v=c1v1+c2v2++cnvn:In other words, picking a basis for a vector space allows us to give coordinates for points.This will allow us to give matrices for linear transformations of vector spaces besidesRn.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeThe matrix of a linear transformation

Denition

LetVandWbe vector spaces withorderedbases

B=fv1;v2;:::;vngandC=fw1;w2;:::;wmg,

respectively, and letT:V!Wbe a linear transformation. Thematrix representation ofTrelative to the basesB andCis

A= [aij]

where T(vj) =a1jw1+a2jw2++amjwm:In other words,Ais the matrix whosej-th column isT(vj), expressed in coordinates usingfw1;:::;wmg.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeExample

LetT:P1!P2be the linear transformation dened by

T(a+bx) = (2a3b) + (b5a)x+ (a+b)x2:

Use basesf1;xgforP1andf1;x;x2gforP2to give a matrix representation ofT.

We have

T(1) = 25x+x2andT(x) =3 +x+x2;

so A 1=2 423
5 1 1 13 5 Now use the basesf1;x+ 5gforP1andf1;1 +x;1 +x2gfor P 2.

We have

T(1) = 25x+x2= 6(1)5(1 +x) + (1 +x2)

and

T(x+ 5) = 724x+ 6x2

= 25(1)24(1 +x) + 6(1 +x2); so A 1=2 423
5 1 1 13 5A 2=2 46 25
524
1 63 5

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeComposition of linear transformations

Denition

LetT1:U!VandT2:V!Wbe linear transformations.

Theircompositionis the linear transformationT2T1dened by (T2T1)(u) =T2(T1(u)):Theorem

LetT1andT2be as above, and letB,C, andDbe ordered

bases forU,V, andW, respectively. IfI A1is the matrix representation forT1relative toBandC,I A2is the matrix representation forT2relative toCandD,I A21is the matrix representation forT2T1relative toB andD,thenA21=A2A1.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeThe inverse of a linear transformation

Denition

IfT:V!Wis a linear transformation, itsinverse(if it exists) is a linear transformationT1:W!Vsuch that

T1T(v) =vandTT1(w) =w

for allv2Vandw2W.Theorem LetTbe as above and letAbe the matrix representation ofT relative to basesBandCforVandW, respectively.Thas an inverse transformation if and only ifAis invertible and, if so, T

1is the linear transformation with matrixA1relative toC

andB.

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeExample

LetT:P2!P2be dened by

T(a+bx+cx2) = (3ab+c) + (ac)x+ (4b+c)x2:Using the basisf1;x;x2gforP2, the matrix representation for Tis A=2 431 1
1 01

0 4 13

5 :This matrix is invertible and A 1=117 2

44 5 1

1 3 4

412 13

5

Thus,T1is given by

T

1(a+bx+cx2) =4a+5b+c17

+a+3b+4c17 x+4a12b+c17 x2:

Linear Trans-

formations

Math 240

Linear Trans-

formations

Transformationsof Euclideanspace

Kernel and

Range

The matrix of

a linear trans.

Composition of

linear trans.

Kernel and

RangeKernel and Range

Theorem

LetT:V!Wbe a linear transformation andAbe a matrix representation ofTrelative to some bases forVandW.I

Ker(T) =fc1v1++cnvn2V: (c1;:::;cn)2

nullspace(A)g,I

Rng(T) =fc1w1++cmwm2W: (c1;:::;cm)2

colspace(A)g.quotesdbs_dbs20.pdfusesText_26