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Chapter 7
Linear TransformationsPo-Ning Chen, Professor
Department of Electrical Engineering
National Chiao Tung University
Hsin Chu, Taiwan 30050, R.O.C.
7.1 The idea of a linear transformation
7-1AtransformationTis simply a mapping fromVtoW.
It is sometimes denoted asT:V?→W.
A transformation is linear if
1.T(v 1 W T(v 2 )=T(v 1 V v 22.T(c·
V v)=c· W T(v) where "+ W V "and"· W V " denote some general "additions" and "mul- tiplications" defined overWandV, respectively.VandWare usually vector spacesVandW.
For simplicity, we will drop the subscripts in "+" and "." (in case there is no ambiguity in these operations).7.1 The idea of a linear transformation
7-2Important notes on linear transformation
A line segment will be transformed to a line segment. T(a 1 v 1 +(1-a 1 )v 2 )=a 1 T(v 1 )+(1-a 1 )T(v 2 )=a 1 w 1 +(1-a 1 )w 2 Hence, a triangle will be transformed into a triangle.0inVwill be transformed to0inW.
T(0)=T(0·v)=0·T(v)=0.
Note again that the0inVand the0inWmay be different. For example, T ?0 0 0? =?0 0?7.1 Kernel
7-3 Definition (Kernel):Thekernelof a transformationTis the set of allvsuch thatT(v)=0.
The concept of "kernel" becomes more evidently important when the transfor- mationTis linear. For a linear transformation, the number of elements in the setK(w)?{v:T(v)=w}
is a constant, independent ofw.Proof:
-Suppose distinctv 1 ,v 2 ,v 3 ,...,v k satisfyingT(v i k,wherek=|K(w)|is the size of the setK(w). Then, either|K(w)|≥k or|K(w)|= 0 because for a givenvsatisfyingT(v)=w,wehaveT(v+v
i -v 1 )=T(v)+T(v i )-T(v 1 -Since we can interchange the role ofwandw, we conclude that|K(w)|= |K(w)|if they are positive.?7.1 Kernel
7-4 Note that sinceT(0)=0for a linear transformation,K(0) cannot be empty. So|V| |K(0)| will give the number of elementswinWsuch thatT(v)=wfor somev. Definition (Range):Therangeof a transformationTis the set of allwsuch thatT(v)=wfor somev.
I.e., {w?W:?vsuch thatT(v)=w}.Important fact about linear transformation
A linear transformation from a vector spaceVto a vector spaceWcan always be represented as Av=w by properly selecting the matrixA.So,?
Kernel=Null spaceofA
Range=Column spaceofA
7.1 Problem discussion
7-5 (Problem 16, Section 7.1) SupposeTtransposes every matrixM. Try to find a matrixAwhich givesAM=M T for everyM. Show that no matrixAwill do it. To professors:Is this a linear transformation that doesn"t come from a matrix. Thinking over Problem 16:Define a transformation that maps a matrixM2×2
to its transposeM T . Is this a linear transformation?Solution.
?T(M
1 +M 2 )=(M 1 +M 2 T =M T1 +M T2 =T(M 1 )+T(M 2T(c·M)=(c·M)
T =cM T =c·T(M) hence, it is a linear transformation.?7.1 Problem discussion
7-6There does not exist any matrixA
2×2
satisfyingAM=M TBut there does exist a matrixA
4×4
satisfying A? ??m 1,1 m 1,2 m 2,1 m 2,2 ??m 1,1 m 2,1 m 1,2 m 2,2 So a linear transformation can always be represented asAv4×1
=w4×1
(since the dimension ofMisfour).7.2 The matrix of a linear transformation
7-7For a linear transformation
T:V?→W,
how to find its equivalent matrix representation A m×n v n×1 =w m×1Answer:
Denote the standard basis for vector spaceVby
e 1 ??1 0... 0? ??,e 2 ??0 1... 0? ??,...,e n ??0 0... 1?Then,T(e
i )=Ae i gives ?T(e 1 )T(e 2 )···T(e n )?=A?e 1 e 2···e
n ?=A.7.2 The matrix of a linear transformation
7-8 Example.v(x) = a polynomial ofxof order 3, andT(v)=∂v(x) ∂x.The bases forv(x)are1,x,x
2 ,x 3 . Or in vector forms,? ??1 0 0 0? ??0 1 0 0? ??0 0 1 0? ??0 0 0 1?So,A=?T(1)T(x)T(x
2 )T(x 3 )?=?012x3x 2Or in matrix form,A=?
?0100 0020 0003?7.2 The matrix of a linear transformation
7-9Hence, ifv(x)=1+2x+x
3 ??1 2 0 1? ??,then ∂v(x) ∂x=A? ??1 2 0 1? ?2 0 3? =2+3x 27.2 The matrix of a linear transformation
7-10For a linear transformation
T:V?→W,
how to find its equivalent matrix representation A m×n v n×1 =w m×1 (by the bases other thane 1 ,e 2 ,...,e nAnswer:
Denote a basis for vector spaceVbyv
1 ,v 2 ,...,v nDenote a basis for vector spaceWbyw
1 ,w 2 ,...,w mSuppose that
T(v i )=b 1,i w 1 +b 2,i w 2 +···+b m,i w mThen,T(v
i )=Av i gives A ?v 1 v 2···v
n ?=?T(v 1 )T(v 2 )···T(v n )?=?w 1···w
m ???b 1,1 b 1,2···b
1,m b 2,1 b 2,2···b
2,m b m,1 b m,2···b
m,m7.2 The matrix of a linear transformation
7-11Hence,
A=?T(v
1 )T(v 2 )···T(v n )??v 1 v 2···v
n -1 =?w 1···w
m ??b 1,1 b 1,2···b
1,m b 2,1 b 2,2···b
2,m b m,1 b m,2···b
m,m ???v 1 v 2···v
n -1 Example (Example 6 in the textbook):Tprojects a vector in? 2 onto the line passing via (0,0) and (1,1). Find the projection matrixA.Solution 1:
A=?
T??1 0?? T??0 1??? 12121
21
2
7.2 The matrix of a linear transformation
7-12Solution 2:
Choosev
1 1 212 andv 2 1 21
2