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Chapter 2 – Linear Transformations and Matrices Per-Olof Persson [email protected] 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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    Addition and Scalar Multiplication

    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.