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Chapter 2 – Linear Transformations and Matrices Per-Olof Persson Department of Mathematics University of California, Berkeley Math 110 Linear Algebra

Teoremas Cap.2

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Chapter 2 Linear Transformations and Matrices

Per-Olof [email protected]

Department of Mathematics

University of California, Berkeley

Math 110 Linear Algebra

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Linear Transformations

Definition

We call a function T: V W a linear transformation from V to Wif, for all x, yV and cF, we have

(a) T(x + y) =T(x) +T(y) and

(b) T(cx) =cT(x)

1 If T is linear, then T(0) = 0

2 T is linear T(cx + y) =cT(x) +T(y) x, yV, c F

3 If T is linear, then T(x y) =T(x) T(y) x, yV

4 T is linear forx1, . . . , xnV and a1, . . . , an F,

T (

ni=1 aixi) =

ni=1 aiT(xi)

Special linear transformations

The identity transformation IV :VV: IV(x) =x, xV

The zero transformation T0:V W: T0(x) = 0 xV

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Null Space and Range

Definition

For linear T:V W, the null space(orkernel) N(T) of T is theset of all xV such that T(x) = 0: N(T) ={xV:T(x) = 0}The range(or image) R(T) of T is the subset of W consisting ofall images of vectors in V: R(T) ={T(x) :xV}

Theorem 2.1

For vector spaces V, W and linear T:V W, N(T) and R(T) aresubspaces of V and W, respectively.

Theorem 2.2For vector spaces V, W and linear T:V W, if={v1, . . . , vn}is a basis for V, then

R(T) = span(T()) = span({T(v1), . . . , T(vn)})

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Nullity and Rank

Definition

For vector spaces V, W and linear T:V W, if N(T) and R(T)are finite-dimensional, the nullityand the rankof T are thedimensions of N(T) and R(T), respectively.

Theorem 2.3 (Dimension Theorem)

For vector spaces V, W and linear T:V W, if V isfinite-dimensional then

nullity(T) + rank(T) = dim(V)

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Properties of Linear Transformations

Theorem 2.4

For vector spaces V, W and linear T:V W, T is one-to-one ifand only if N(T) ={0}.

Theorem 2.5For vector spaces V, W of equal (finite) dimension and linearT:V W, the following are equivalent:

(a) T is one-to-one

(b) T is onto(c) rank(T) = dim(V)

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Linear Transformations and Bases

Theorem 2.6

Let V, W be vector spaces overF and{v1, . . . , vn} a basis for V.Forw1, . . . , wn in W, there exist exactly one linear transformation

T:V W such that T(vi) =wi for= 1, . . . , n.

Corollary

Suppose{v1, . . . , vn} is a finite basis for V, then if U, T: VW

are linear and U(vi) =T(vi) fori= 1, . . . , n, then U=T.

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Coordinate Vectors

Definition

For a finite-dimensional vector space V, an ordered basisfor V is abasis for V with a specific order. In other words, it is a finitesequence of linearly independent vectors in V that generates V.

Definition

Let ={u1, . . . , un}be an ordered basis for V, and for xV leta1, . . . , an be the unique scalars such that

x=n

i=1aiui.

The coordinate vector ofxrelative to is

[x] =

a1...

an

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Matrix Representations

Definition

Suppose V, W are finite-dimensional vector spaces with orderedbases ={v1, . . . , vn}, ={w1, . . . , wm}. For linear T: V W,there are unique scalars aij F such that

T(vj) =

mi=1

aijwi for1 jn.

The m n matrix A defined by Aij =aij is the matrixrepresentation of T in the ordered basesand, written

A= [T]

. If V=W and =, then A= [T] .

Note that the jth column ofA is [T(vj)], and if[U] = [T]

for

linear U:V W, then U=T.

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Definition

Let T, U:V W be arbitrary functions of vector spaces V, W over

F. Then T+U, aT:V W are defined by(T+U)(x) =T(x) +U(x) and (aT)(x) =aT(x), respectively, forall x V and aF.

Theorem 2.7

With the operations defined above, for vector spaces V, W overFand linear T, U:V W:

(a) aT+U is linear for alla F

(b) The collection of all linear transformations from V to W is a

vector space overF

Definition

For vector spaces V, W over F, the vector space of all lineartransformations from V into W is denoted by L(V, W), or justL(V) if V=W.

M i R i

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Matrix Representations

Theorem 2.8

For finite-dimensional vector spaces V, W with ordered bases, ,

and linear transformations T, U: VW:(a) [T+U] = [T]

+ [U]

(b) [aT] =a[T] for all scalarsa

C i i f Li T f i

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Composition of Linear Transformations

Theorem 2.9

Let V, W, Z be vector spaces over a fieldF, and T : VW,U : WZ be linear. Then UT : VZ is linear.

Theorem 2.10Let V be a vector space and T, U1, U2 L(V). Then

(a) T(U1+U2) =TU1+TU2 and(U1+U2)T=U1T+U2T

(b) T(U1U2) = (TU1)U2

(c) TI=IT= T(d) a(U1U2) = (aU1)U2=U1(aU2) for all scalarsa

M i M l i li i

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Matrix Multiplication

Let T : V W, U : W Z, be linear, = {v1, . . . , vn},

={w1, . . . , wm}, ={z1, . . . , zp} ordered bases for U, W, Z,and A= [U], B= [T]

. Consider [UT]

:

(UT)(vj) =U(T(vj)) =U

m

k=1Bkjwk

=

m

k=1BkjU(wk)

=mk=1

Bkj

pi=1

Aikzi

=

pi=1

mk=1

AikBkj

zi

Definition

Let A,B be m n, n p matrices. The productAB is the m pmatrix with

(AB)ij =n

k=1AikBkj , for1 im, 1 j p

M t i M lti li ti

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Matrix Multiplication

Theorem 2.11

Let V, W, Z be finite-dimensional vector spaces with ordered bases,, , and T : VW, U : WZ be linear. Then

[UT]= [U][T]

Corollary

Let V be a finite-dimensional vector space with ordered basis,and T, U L(V). Then [UT] = [U][T].

Definition

The Kronecker delta is defined by ij = 1 ifi=j and ij = 0 ifi=j . The n n identity matrixIn is defined by (In)ij =ij .

P ti

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Properties

Theorem 2.12

LetAbem nmatrix, B,C ben p matrices, andD, Ebeq mmatrices. Then

(a) A(B+ C) =AB+ AC and(D+ E)A=DA + EA

(b) a(AB) = (aA)B =A(aB) for any scalara

(c) ImA= A =AIn

(d) If V is an n-dimensional vector space with ordered basis,then [IV] =In

Corollary

LetAbem nmatrix, B1, . . . , Bk ben p matrices, C1, . . . , C kbeq mmatrices, anda1, . . . , ak be scalars. Then

A k

i=1

aiBi =k

i=1

aiABi and k

i=1

aiCiA=k

i=1

aiCiA

Properties

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Properties

Theorem 2.13LetAbem nmatrix andB ben p matrix, anduj, vj thejthcolumns ofAB,B. Then

(a) uj =Avj

(b) vj =Bej

Theorem 2.14

Let V, W be finite-dimensional vector spaces with ordered bases

, , and T : VW be linear. Then foru V:

[T(u)]= [T] [u]

Left multiplication Transformations

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Left-multiplication Transformations

Definition

Let Abe m nmatrix. The left-multiplication transformation LAis the mapping LA:Fn Fm defined by LA(x) =Axfor eachcolumn vector x Fn.

Theorem 2.15

LetAbem nmatrix, then LA:Fn

Fm

is linear, and ifB ism nmatrix and, are standard ordered bases for Fn, Fm, then:

(a) [LA] =A

(b) LA =LB if and only ifA= B

(c) LA+B =LA+LB and LaA=aLA for alla F(d) For linear T: Fn Fm, there exists a uniquem n matrixC

such that T= LC, andC= [T]

(e) IfE is an n p matrix, then LAE=LALE

(f) Ifm= n then LIn

=IFn

Associativity of Matrix Multiplication

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Associativity of Matrix Multiplication

Theorem 2.16

LetA, B, Cbe matrices such thatA(BC) is defined. Then(AB)C is also defined andA(BC) = (AB)C.

Inverse of Linear Transformations

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Inverse of Linear Transformations

Definition

Let V, W be vector spaces and T : V W be linear. A function U: W V is an inverseof T if TU=IW and UT=IV. If T has aninverse, it is invertibleand the inverse T1 is unique.

For invertible T,U:

1 (TU)1 =U1T1

2 (T1)1 =T (so T1 is invertible)

3 If V,W have equal dimensions, linear T : V W is invertibleif and only ifrank(T) = dim(V)

Theorem 2.17

For vector spaces V,W and linear and invertible T : VW,T1 : WV is linear.

Inverses

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Inverses

DefinitionAn n nmatrix A is invertible if there exists an n nmatrix Bsuch that AB=BA= I.

Lemma

For invertible and linear T from V to W, V is finite-dimensional ifand only if W is finite-dimensional. Then dim(V) = dim(W).

Theorem 2.18

Let V,W be finite-dimensional vector spaces with ordered bases, , and T : VW be linear. Then T is invertible if and only if[T] is invertible, and [T

1] = ([T])1.

Inverses

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Inverses

Corollary 1

For finite-dimensional vector space V with ordered basisandlinear T : VV, T is invertible if and only if[T] is invertible,

and[T1

] = ([T ])1

.

Corollary 2

An n nmatrixAis invertible if and only if LA is invertible, and(L

A)1 =L

A1 .

Isomorphisms

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Isomorphisms

DefinitionLet V,W be vector spaces. V is isomorphicto W if there exists alinear transformation T : V W that is invertible. Such a T is anisomorphismfrom V onto W.

Theorem 2.19

For finite-dimensional vector spaces V,W, V is isomorphic to W ifand only ifdim(V) = dim(W).

CorollaryA vector space V overF is isomorphic to Fn if and only ifdim(V) =n.

Linear Transformations and Matrices

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Linear Transformations and Matrices

Theorem 2.20

Let V,W be finite-dimensional vector spaces overFof dimensions

n,mwith ordered bases, . Then the function :L(V, W) Mmn(F), defined by(T) = [T]

forT L(V, W), is an isomorphism.

Corollary

For finite-dimensional vector spaces V,W of dimensionsn,m,

L(V, W) is finite-dimensional of dimension mn.

The Standard Representation

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The Standard Representation

Definition

Let be an ordered basis for an n-dimensional vector space V overthe field F. The standard representation of V with respect to is

the function :VFn

defined by (x) = [x] for each xV.

Theorem 2.21

For any finite-dimensional vector space V with ordered basis, is an isomorphism.

The Change of Coordinate Matrix

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The Change of Coordinate Matrix

Theorem 2.22Letand be ordered bases for a finite-dimensional vector spaceV, and letQ= [IV]

. Then

(a) Q is invertible

(b) For anyvV, [v] =Q[v]

Q= [IV] is called a change of coordinate matrix, and we say that

Qchanges-coordinates into-coordinates.

Note that ifQchanges from into coordinates, then Q1

changes from into coordinates.

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Linear Functionals

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A linear functionalon a vector space V is a linear transformationfrom V into its field of scalars F.

Example

Let V be the continuous real-valued functions on [0, 2]. For a fix

gV, a linear functional h: VR is given by

h(x) = 1

2

2

0

x(t)g(t) dt

Example

Let V=Mnn(F), then f :V F with f(A) = tr(A) is a linearfunctional.

Coordinate Functions

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Example

Let ={x1, . . . , xn} be a basis for a finite-dimensional vectorspace V. Define fi(x) =ai, where

[x] =a1...

an

is the coordinate vector ofx relative to . Then fi is a linear

functional on V called the ith coordinate function with respect tothe basis. Note that fi(xj) =ij .

Dual Spaces

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p

Definition

For a vector space V over F, the dual spaceof V is the vectorspace V =L(V, F).

Note that for finite-dimensional V,

dim(V) = dim(L(V, F)) = dim(V) dim(F) = dim(V)

so V and V are isomorphic. Also, the double dualV of V is thedual of V.

Dual Bases

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Theorem 2.24

Let={x1, . . . , xn} be an ordered basis for finite-dimensional

vector space V, and let fi be theith coordinate function w.r.t. ,and ={f1, . . . , fn}. Then

is an ordered basis for V and forany fV,

f=n

i=1f(xi)fi.

Definition

The ordered basis ={f1, . . . , fn}of V that satisfies fi(xj) =ij

is called the dual basisof.

Theorem 2.25

Let V, W be finite-dimensional vector spaces overFwith orderedbases, . For any linear T : VW, the mapping Tt :W V

defined by Tt(g) =gT for all gW is linear with the property

[Tt

]

= ([T)

t

.

Double Dual Isomorphism

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For a vector xV, define x: V F by x(f) =f(x) for every

fV

. Note that x is a linear functional on V

, so xV

.

Lemma

For finite-dimensional vector space V andxV, ifx(f) = 0 for allfV, then x= 0.

Theorem 2.26

Let V be a finite-dimensional vector space, and define:VV

by(x) = x. Then is an isomorphism.

Corollary

For finite-dimensional V with dual space V, every ordered basis forV is the dual basis for some basis for V.

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