Solution Manual for Linear Algebra, 5th Edition

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SOLUTIONSMANUALLINEARALGEBRAFIFTHEDITIONStephen H. FriedbergArnold J. InselLawrence E. SpenceIllinois State University

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Contents1Vector Spaces11.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.3Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.4Linear Combinations and Systems of Linear Equations . . . . . . . . . . . . . . . . .21.5Linear Dependence and Linear Independence. . . . . . . . . . . . . . . . . . . . . .21.6Bases and Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Linear Transformations and Matrices42.1Linear Transformations, Null Spaces, and Ranges . . . . . . . . . . . . . . . . . . . .42.2The Matrix Representation of a Linear Transformation. . . . . . . . . . . . . . . .42.3Composition of Linear Transformations and Matrix Multiplication. . . . . . . . . .52.4Invertibility and Isomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.5The Change of Coordinate Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . .62.6Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.7Homogeneous Linear Differential Equations with Constant Coefficients . . . . . . . .63Elementary Matrix Operations and Systems of Linear Equations83.1Elementary Matrix Operations and Elementary Matrices. . . . . . . . . . . . . . .83.2The Rank of a Matrix and Matrix Inverses. . . . . . . . . . . . . . . . . . . . . . .83.3Systems of Linear Equations—Theoretical Aspects . . . . . . . . . . . . . . . . . . .93.4Systems of Linear Equations—Computational Aspects . . . . . . . . . . . . . . . . .94Determinants114.1Determinants of Order 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114.2Determinants of Ordern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114.3Properties of Determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114.4Summary–Important Facts about Determinants. . . . . . . . . . . . . . . . . . . .124.5A Characterization of the Determinant. . . . . . . . . . . . . . . . . . . . . . . . .125Diagonalization135.1Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135.2Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145.3Matrix Limits and Markov Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . .145.4Invariant Subspaces and the Cayley-Hamilton Theorem. . . . . . . . . . . . . . . .15iii

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Table of Contents6Inner Product Spaces166.1Inner Products and Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166.2The Gram-Schmidt Orthogonalization Process and Orthogonal Complements . . . .166.3The Adjoint of a Linear Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . .176.4Normal and Self-Adjoint Operators. . . . . . . . . . . . . . . . . . . . . . . . . . .176.5Unitary and Orthogonal Operators and Their Matrices. . . . . . . . . . . . . . . .186.6Orthogonal Projections and the Spectral Theorem. . . . . . . . . . . . . . . . . . .186.7The Singular Value Decomposition and the Pseudoinverse . . . . . . . . . . . . . . .196.8Bilinear and Quadratic Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206.10Conditioning and the Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . .206.11The Geometry of Orthogonal Operators . . . . . . . . . . . . . . . . . . . . . . . . .207Canonical Forms217.1Jordan Canonical Form I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217.2Jordan Canonical Form II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217.3The Minimal Polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227.4Rational Canonical Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22iv

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1Vector Spaces1.1INTRODUCTION2.(b)x= (2,4,0) +t(−5,−10,0)(d)x= (−2,−1,5) +t(5,10,2)3.(b)x= (3,−6,7) +s(−5,6,−11) +t(2,−3,−9)(d)x= (1,1,1) +s(4,4,4) +t(−7,3,1)4.(0,0)1.2VECTOR SPACES2.0000000000004.(b)1−13−538(d)30−20−1510−5−40(f )−x3+ 7x2+ 4(h)3x5−6x3+ 12x+ 65.831300300+914300110=174560041016.Yes18.No, (VS 1) fails.19.No, (VS 8) fails.1.3SUBSPACES2.(b)0384−67(d)102−50−47−836The trace is 12.(f )−2750114−6(h)−40601−36−35The trace is 2.8.(b)No(d)Yes(f )No9.W1∩W3={(0,0,0)},W1∩W4=W1,W3∩W4={(a1, a2, a3)∈R3:a1=−11a3anda2=−3a3}1

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Chapter 1Vector Spaces1.4LINEAR COMBINATIONS AND SYSTEMS OF LINEAR EQUATIONS2.(b)(−2,−4,−3)(d){x3(−8,3,1,0) + (−16,9,0,2):x3∈R}(f )(3,4,−2)3.(a)(−2,0,3) = 4(1,3,0)−3(2,4,−1)(b)(1,2,−3) = 5(−3,2,1) + 8(2,−1,−1)(d)No(f )(−2,2,2) = 4(1,2,−1) + 2(−3,−3,3)4.(a)x3−3x+ 5 = 3(x3+ 2x2−x+ 1)−2(x3+ 3x2−1)(b)No(c)−2x3−11x2+ 3x+ 2 = 4(x3−2x2+ 3x−1)−3(2x3+x2+ 3x−2)(d)x3+x2+ 2x+ 13 =−2(2x3−3x2+ 4x+ 1) + 5(x3−x2+ 2x+ 3)(f )No5.(b)No(d)Yes(f )No(h)No11.The span of{x}is{0}ifx=0and is the line through the origin ofR3in the direction ofxifx6=0.17.ifWis finite1.5LINEAR DEPENDENCE AND LINEAR INDEPENDENCE2.(b)Linearly independent(d)Linearly dependent(f )Linearly independent(h)Linearly independent(j)Linearly dependent10.(1,0,0), (0,1,0), (1,1,0)1.6BASES AND DIMENSION2.(b)Not a basis(d)Basis3.(b)Basis(d)Basis4.No, dim(P3(R)) = 4.5.No, dim(R3) = 3.8.{u1, u3, u5, u7}10.(b)12−3x(d)2x3−x2−6x+ 1514.{(0,1,0,0,0), (0,0,0,0,1), (1,0,1,0,0), (1,0,0,1,0)}and{(−1,0,0,0,1), (0,1,1,1,0)}; dim(W1) = 4 and dim(W2) = 2.16.dim(W) =12n(n+ 1)2

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1.6Bases and Dimension18.Letσjbe the sequence such thatσj(i) ={1i=j0i6=j.Then{σj:j= 1,2, . . .}is a basis for the vector space in Example 5 of Section 1.2.22.W1⊆W223.(a)v∈W1(b)dim(W2) = dim(W1) + 125.mn27.Ifnis even, then dim(W1) = dim(W2) =n2 ; and ifnis odd,then dim(W1) =n+ 12and dim(W2) =n−12.32.(a)TakeW1=R3andW2= span({e1}).(b)TakeW1= span({e1, e2}) andW2= span({e3}).(c)TakeW1= span({e1, e2}) andW2= span({e2, e3}).35.(b)dim(V) = dim(W) + dim(V/W)3

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2Linear Transformationsand Matrices2.1LINEAR TRANSFORMATIONS, NULL SPACES, AND RANGES3.The nullity is 0, and the rank is 2. ThusTis one-to-one, but not onto.6.The nullity isn−1, and the rank is 1. ThusTis not one-to-one unlessn= 1, andTis notonto unlessn= 1.11.T(8,11) = (5,−3,16)18.T(a, b) = (b,0).N(T) = span{(1,0)}=R(T).19.LetV=W=R, and defineT=IandU= 2I.23.All ofR3or a plane inR3through the origin25.(a)T(a, b) = (0, b)(b)T(a, b) = (0, b−a)26.(b)T(a, b, c) = (0,0, c)28.(b)See (a) and (b) of Exercise 25.32.(c)LetV=P(F) andW= span({1}). DefineTfirst on the standard basis ofVbyT(1) =T(x) =0, andT(xk) =xk−1fork≥2. Now extendTto a linear transformation fromVtoV. ThenN(T) = span({1, x}), andR(T) = span({xk:k≥1}). SoV=R(T)⊕W, butW6=N(T).2.2THE MATRIX REPRESENTATION OF A LINEAR TRANSFORMATION2.(b)(23−1101)(e)10. . .010. . .0.........10. . .04.1100000201005.(c)(1 0 0 1)(d)(1 2 4)(f )3−61(g)(a)4

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