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Linear Trans-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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 y00+y= 0
is equivalent to nding functionsysuch thatT(y) = 0, whereTis dened as
T(y) =y00+y:
Linear Trans-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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 transformationfromVtoWif 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 IT:Rn!Rmdened byT(x) =Ax, whereAis an
mnmatrixIT: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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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.IIfy2Ck(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):IIfy2Ck(I)andc2R, then
T(cy) = (cy)00+ (cy) =cy00+cy=c(y00+y) =cT(y):
Linear Trans-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe matrix of
a linear trans.Composition of
linear trans.Kernel and
RangeLinear transformations fromRntoRmLetAbe anmnmatrix with real entries and deneT:Rn!RmbyT(x) =Ax. Verify thatTis a linear
transformation.IIfxis ann1column vector thenAxis anm1
column vector.IT(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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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, whereA=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 byT(x1;x2;x3;x4) = (2x1+ 3x2+x4;5x1+ 9x3x4;
4x1+ 2x2x3+ 7x4):
Linear Trans-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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 denotedKer(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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe matrix of
a linear trans.Composition of
linear trans.Kernel and
RangeRank-Nullity revisited
SupposeTis the matrix transformation withmnmatrixA.We know IKer(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:TheoremIfT:V!Wis a linear transformation andVis
nite-dimensional, then dim(Ker(T)) + dim(Rng(T)) = dim(V):Linear Trans-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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 andCisA= [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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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 4235 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)
andT(x+ 5) = 724x+ 6x2
= 25(1)24(1 +x) + 6(1 +x2); so A 1=2 4235 1 1 13 5A 2=2 46 25
524
1 63 5
Linear Trans-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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)):TheoremLetT1andT2be 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-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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 thatT1T(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, T1is the linear transformation with matrixA1relative toC
andB.Linear Trans-
formationsMath 240
Linear Trans-
formationsTransformationsof Euclideanspace
Kernel and
RangeThe 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 11 01