Lecture 14 Linear Transformation

Last Time- Mathematical Models and Least Square Analysis - Inner Product Space Applications- Introduction to Linear Transformations

Elementary Linear AlgebraR. Larsen et al. (5 Edition) TKUEE翁慶昌 -NTUEE SCC_01_2008

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Lecture 14: Linear Transformation

Today The Kernel and Range of a Linear Transformation Matrices for Linear Transformations Transition Matrix and SimilarityReading Assignment: Secs 6.2-6.4

Next Time Applications of Linear Transformations Eigenvalues and Eigenvectors DiagonalizationReading Assignment: Secs 6.5-7.2

What Have You Actually Learned about LSPs So Far?

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Keywords in Section 6.1:

function: 函數 domain: 論域 codomain: 對應論域 image of v under T: 在 T映射下 v的像 range of T: T的值域 preimage of w: w的反像 linear transformation: 線性轉換 linear operator: 線性運算子 zero transformation: 零轉換 identity transformation: 相等轉換

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Today The Kernel and Range of a Linear Transformation Matrices for Linear Transformations Transition Matrix and Similarity

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6.2 The Kernel and Range of a Linear Transformation Kernel of a linear transformation T:

Let be a linear transformationWVT :

Then the set of all vectors v in V that satisfy is called the kernel of T and is denoted by ker(T).

0)( vT

} ,0)(|{)ker( VTT vvv

Ex 1: (Finding the kernel of a linear transformation) ):( )( 3223 MMTAAT T

Sol:

000000

)ker(T

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Ex 2: (The kernel of the zero and identity transformations)

(a) T(v)=0 (the zero transformation )WVT :

VT )ker(

(b) T(v)=v (the identity transformation )VVT :

}{)ker( 0T

Ex 3: (Finding the kernel of a linear transformation) ):( )0,,(),,( 33 RRTyxzyxT

?)ker( TSol:

}number real a is |),0,0{()ker( zzT

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Ex 5: (Finding the kernel of a linear transformation)

):( 321211)( 23

3

2

1

RRTxxx

AT

xx

?)ker( T

Sol:}),,( ),0,0(),,(|),,{()ker( 3

321321321 RxxxxxxxTxxxT

)0,0(),,( 321 xxxT

00

321211

3

2

1

xxx

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11

1

3

2

1

ttt

t

xxx

)}1,1,1span{( }number real a is |)1,1,1({)ker(

ttT

01100101

03210211 .. EJG

Thm 6.3: (The kernel is a subspace of V)

The kernel of a linear transformation is a subspace of the domain V.

WVT :

)16. Theorem( 0)0( TPf:VT ofsubset nonempty a is )ker(

then. of kernel in the vectorsbe and Let Tvu

000)()()( vuvu TTT

00)()( ccTcT uu )ker(Tc u

)ker(T vu

. of subspace a is )ker(Thus, VT Note:

The kernel of T is sometimes called the nullspace of T. 14 - 10

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Ex 6: (Finding a basis for the kernel)

82000102010131211021

and Rin is where,)(by defined be :Let 545

A

ATRRT xxx

Find a basis for ker(T) as a subspace of R5.

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Sol:

000000041000020110010201

082000010201001312011021

0

.. EJG

A

s t

14

021

00112

4

22

5

4

3

2

1

ts

tt

ststs

xxxxx

x

TB of kernel for the basis one:)1 ,4 ,0 ,2 ,1(),0 ,0 ,1 ,1 ,2(

Corollary to Thm 6.3:

0 of spacesolution the toequal is T of kernel Then the)(by given L.T thebe :Let

xxx

AATRRT mn

) of subspace( ,0|)()(

):on ansformatilinear tr a( )(mm

mn

RRAANSTKer

RRTAT

xxxxx

Range of a linear transformation T:

)(by denoted is and T of range thecalled is Vin vector of images arein W that w vectorsall ofset Then the

L.T. a be :Let

Trange

WVT

}|)({)( VTTrange vv

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. :Tn nsformatiolinear tra a of range The WWV fo ecapsbus a si

Thm 6.4: (The range of T is a subspace of W)

Pf:)1Thm.6.( 0)0( T

WTrange ofsubset nonempty a is )(

TTT of range in the vector be )( and )(Let vu

)()()()( TrangeTTT vuvu

)()()( TrangecTcT uu

),( VVV vuvu

)( VcV uu

.subspace is )( Therefore, WTrange

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Notes:

of subspace is )()1( VTKer

L.T. a is : WVT

Corollary to Thm 6.4:

)()( of spacecolumn the toequal is of range Then the

)(by given L.T. thebe :Let

ACSTrangeAT

ATRRT mn

xx

of subspace is )()2( WTrange

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Ex 7: (Finding a basis for the range of a linear transformation)

82000102010131211021

and is where,)(by defined be :Let 545

A

RATRRT xxx

Find a basis for the range of T.

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Sol:

BA EJG

0000 04100 02011 01020 1

82000102010131211021

..

54321 ccccc 54321 wwwww

)( , ,

)( , ,

421

421

ACScccBCSwww

for basis a is for basis a is

T of range for the basis a is )2 ,0 ,1 ,1( ),0 ,0 ,1 ,2( ),0 ,1 ,2 ,1(

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Rank of a linear transformation T:V→W:TTrank of range theofdimension the)(

Nullity of a linear transformation T:V→W:TTnullity of kernel theofdimension the)(

Note:

)()()()(

then,)(by given L.T. thebe :Let

AnullityTnullityArankTrank

ATRRT mn

xx

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then. space vector a into space vector ldimensiona-nan form L.T. a be :Let

WVWVT

Thm 6.5: (Sum of rank and nullity)

Pf:AmatrixnmT an by drepresente is Let

) ofdomain dim() of kerneldim() of rangedim()()(

TTTnTnullityTrank

rArank )( Assume

rArankATTrank

)( ) of spacecolumn dim() of rangedim()((1)

nrnrTnullityTrank )()()(rn

ATTnullity

) of spacesolution dim() of kerneldim()()2(

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Ex 8: (Finding the rank and nullity of a linear transformation)

000110201

by define : L.T. theofnullity andrank theFind 33

A

RRT

Sol:

123)() ofdomain dim()(2)()(

TrankTTnullityArankTrank

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Ex 9: (Finding the rank and nullity of a linear transformation)

}0{)( if ofrank theFind )(4 is ofnullity theif ofrank theFind )(

2 is range theof dimension theif of kernel theofdimension theFind )(

n.nsformatiolinear tra a be :Let 75

TKerTcTTb

TaRRT

Sol:

325) of rangedim() of kerneldim( 5) ofdomain dim()(

TnTTa

145)()()( TnullitynTrankb

505)()()( TnullitynTrankc

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vector.single a of consists range in the every wof preimage theif one-to-one called is :function A WVT

One-to-one:

. that implies )()( inV, vandu allfor iff one-to-one is

vuvu

TTT

one-to-one not one-to-one

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in preimage a has in element every if onto be tosaid is :function A

VWVT

w

Onto:

(T is onto W when W is equal to the range of T.)

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Thm 6.6: (One-to-one linear transformation)

}0{)( iff 1-1 is TThen L.T. a be :Let

TKerWVT

Pf:1-1 is Suppose T

0 :solution oneonly havecan 0)(Then vvT

}0{)( i.e. TKer

)()( and }0{)( Suppose vTuTTKer

0)()()( vTuTvuTL.T. a is T

0)( vuTKervu1-1 is T

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Ex 10: (One-to-one and not one-to-one linear transformation)

one.-to-one is )(by given : L.T. The )( T

mnnm AATMMTa

matrix. zero only the of consists kernel its Because nm

one.-to-onenot is :ation transformzero The )( 33 RRTb . of all is kernel its Because 3R

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Thm 6.7: (Onto linear transformation)

. ofdimension the toequal is ofrank theiff onto is Then l.dimensiona finite is whereL.T., a be :Let

WTTWWVT

Thm 6.8: (One-to-one and onto linear transformation)

onto. isit ifonly and if one-to-one is Then .dimension ofboth and spaceor with vectL.T. a be :Let

TnWVWVT

Pf:0))(dim( and }0{)( then one,-to-one is If TKerTKerT

)dim())(dim())(dim( WnTKernTrange onto. is ly,Consequent T

0) of rangedim())(dim( nnTnTKerone.-to-one is Therefore,T

nWTT )dim() of rangedim( then onto, is If

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Ex 11:

neither.or onto, one,-to-one is whether determine and ofrank andnullity theFind ,)(by given is : L.T. The

TTATRRT mn xx

100110021

)( Aa

001021

)( Ab

110

021)( Ac

000110021

)( Ad

Sol:T:Rn→Rm dim(domain

of T) rank(T) nullity(T) 1-1 onto

(a)T:R3→R3 3 3 0 Yes Yes(b)T:R2→R3 2 2 0 Yes No(c)T:R3→R2 3 2 1 No Yes(d)T:R3→R3 3 2 1 No No

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Isomorphism:

other.each toisomorphic be tosaid are and then , to from misomorphisan exists theresuch that

spaces vector are and if Moreover, m.isomorphisan called is onto and one toone is that :n nsformatiolinear traA

WVWVWV

WVT

Thm 6.9: (Isomorphic spaces and dimension)

Pf:.dimension has where, toisomorphic is that Assume nVWV

onto. and one toone is that : L.T. a exists There WVT one-to-one is T

nnTKerTTTKer

0))(dim() ofdomain dim() of rangedim(0))(dim(

Two finite-dimensional vector space V and W are isomorphic if and only if they are of the same dimension.

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.dimension haveboth and that Assume nWV

onto. is TnWT )dim() of rangedim(

nWV )dim()dim( Thus

. of basis a be , , ,let

and V, of basis a be , , ,Let

21

21

Wwwwvvv

n

n

nnvcvcvcV

2211

as drepresente becan in vector arbitrary an Then v

nnwcwcwcTWVT

2211)(follows. as : L.T. a definecan you and

v

It can be shown that this L.T. is both 1-1 and onto.Thus V and W are isomorphic.

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Ex 12: (Isomorphic vector spaces)

space-4)( 4 Ra

matrices 14 all of space )( 14 Mb

matrices 22 all of space )( 22 Mc

lessor 3 degree of spolynomial all of space )()( 3 xPd

) of subspace}(number real a is ),0 , , , ,{()( 54321 RxxxxxVe i

The following vector spaces are isomorphic to each other.

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Keywords in Section 6.2:

kernel of a linear transformation T: 線性轉換 T的核空間 range of a linear transformation T: 線性轉換 T的值域 rank of a linear transformation T: 線性轉換 T的秩 nullity of a linear transformation T: 線性轉換 T的核次數 one-to-one: 一對一 onto: 映成 isomorphism(one-to-one and onto): 同構 isomorphic space: 同構的空間