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Solution Manual for Linear Algebra, 5th Edition

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    S OLUTIONS M ANUAL L INEAR A LGEBRA F IFTH E DITION Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Illinois State University

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    Contents 1 Vector Spaces 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Linear Combinations and Systems of Linear Equations . . . . . . . . . . . . . . . . . 2 1.5 Linear Dependence and Linear Independence . . . . . . . . . . . . . . . . . . . . . . 2 1.6 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Linear Transformations and Matrices 4 2.1 Linear Transformations, Null Spaces, and Ranges . . . . . . . . . . . . . . . . . . . . 4 2.2 The Matrix Representation of a Linear Transformation . . . . . . . . . . . . . . . . 4 2.3 Composition of Linear Transformations and Matrix Multiplication . . . . . . . . . . 5 2.4 Invertibility and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 The Change of Coordinate Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.7 Homogeneous Linear Differential Equations with Constant Coefficients . . . . . . . . 6 3 Elementary Matrix Operations and Systems of Linear Equations 8 3.1 Elementary Matrix Operations and Elementary Matrices . . . . . . . . . . . . . . . 8 3.2 The Rank of a Matrix and Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Systems of Linear Equations—Theoretical Aspects . . . . . . . . . . . . . . . . . . . 9 3.4 Systems of Linear Equations—Computational Aspects . . . . . . . . . . . . . . . . . 9 4 Determinants 11 4.1 Determinants of Order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Determinants of Order n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.4 Summary–Important Facts about Determinants . . . . . . . . . . . . . . . . . . . . 12 4.5 A Characterization of the Determinant . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Diagonalization 13 5.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3 Matrix Limits and Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . 15 iii

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    Table of Contents 6 Inner Product Spaces 16 6.1 Inner Products and Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements . . . . 16 6.3 The Adjoint of a Linear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.4 Normal and Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.5 Unitary and Orthogonal Operators and Their Matrices . . . . . . . . . . . . . . . . 18 6.6 Orthogonal Projections and the Spectral Theorem . . . . . . . . . . . . . . . . . . . 18 6.7 The Singular Value Decomposition and the Pseudoinverse . . . . . . . . . . . . . . . 19 6.8 Bilinear and Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.10 Conditioning and the Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.11 The Geometry of Orthogonal Operators . . . . . . . . . . . . . . . . . . . . . . . . . 20 7 Canonical Forms 21 7.1 Jordan Canonical Form I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7.2 Jordan Canonical Form II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7.3 The Minimal Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7.4 Rational Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 iv

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    1 Vector Spaces 1.1 INTRODUCTION 2. (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) 1.2 VECTOR SPACES 2.   0 0 0 0 0 0 0 0 0 0 0 0   4. (b)   1 − 1 3 − 5 3 8   (d)   30 − 20 − 15 10 − 5 − 40   (f ) − x 3 + 7 x 2 + 4 (h) 3 x 5 − 6 x 3 + 12 x + 6 5.   8 3 1 3 0 0 3 0 0   +   9 1 4 3 0 0 1 1 0   =   17 4 5 6 0 0 4 1 0   16. Yes 18. No, (VS 1) fails. 19. No, (VS 8) fails. 1.3 SUBSPACES 2. (b)   0 3 8 4 − 6 7   (d)   10 2 − 5 0 − 4 7 − 8 3 6   The trace is 12. (f )     − 2 7 5 0 1 1 4 − 6     (h)   − 4 0 6 0 1 − 3 6 − 3 5   The trace is 2. 8. (b) No (d) Yes (f ) No 9. W 1 ∩ W 3 = { (0 , 0 , 0) } , W 1 ∩ W 4 = W 1 , W 3 ∩ W 4 = { ( a 1 , a 2 , a 3 ) ∈ R 3 : a 1 = − 11 a 3 and a 2 = − 3 a 3 } 1

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    Chapter 1 Vector Spaces 1.4 LINEAR COMBINATIONS AND SYSTEMS OF LINEAR EQUATIONS 2. (b) ( − 2 , − 4 , − 3) (d) { x 3 ( − 8 , 3 , 1 , 0) + ( − 16 , 9 , 0 , 2): x 3 ∈ 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) x 3 − 3 x + 5 = 3( x 3 + 2 x 2 − x + 1) − 2( x 3 + 3 x 2 − 1) (b) No (c) − 2 x 3 − 11 x 2 + 3 x + 2 = 4( x 3 − 2 x 2 + 3 x − 1) − 3(2 x 3 + x 2 + 3 x − 2) (d) x 3 + x 2 + 2 x + 13 = − 2(2 x 3 − 3 x 2 + 4 x + 1) + 5( x 3 − x 2 + 2 x + 3) (f ) No 5. (b) No (d) Yes (f ) No (h) No 11. The span of { x } is { 0 } if x = 0 and is the line through the origin of R 3 in the direction of x if x 6 = 0 . 17. if W is finite 1.5 LINEAR DEPENDENCE AND LINEAR INDEPENDENCE 2. (b) Linearly independent (d) Linearly dependent (f ) Linearly independent (h) Linearly independent (j) Linearly dependent 10. (1 , 0 , 0), (0 , 1 , 0), (1 , 1 , 0) 1.6 BASES AND DIMENSION 2. (b) Not a basis (d) Basis 3. (b) Basis (d) Basis 4. No, dim( P 3 ( R )) = 4. 5. No, dim( R 3 ) = 3. 8. { u 1 , u 3 , u 5 , u 7 } 10. (b) 12 − 3 x (d) 2 x 3 − x 2 − 6 x + 15 14. { (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( W 1 ) = 4 and dim( W 2 ) = 2. 16. dim( W ) = 1 2 n ( n + 1) 2

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    1.6 Bases and Dimension 18. Let σ j be the sequence such that σ j ( i ) = { 1 i = j 0 i 6 = j. Then { σ j : j = 1 , 2 , . . . } is a basis for the vector space in Example 5 of Section 1.2. 22. W 1 ⊆ W 2 23. (a) v ∈ W 1 (b) dim( W 2 ) = dim( W 1 ) + 1 25. mn 27. If n is even, then dim( W 1 ) = dim( W 2 ) = n 2 ; and if n is odd, then dim( W 1 ) = n + 1 2 and dim( W 2 ) = n − 1 2 . 32. (a) Take W 1 = R 3 and W 2 = span( { e 1 } ). (b) Take W 1 = span( { e 1 , e 2 } ) and W 2 = span( { e 3 } ). (c) Take W 1 = span( { e 1 , e 2 } ) and W 2 = span( { e 2 , e 3 } ). 35. (b) dim( V ) = dim( W ) + dim( V / W ) 3

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    2 Linear Transformations and Matrices 2.1 LINEAR TRANSFORMATIONS, NULL SPACES, AND RANGES 3. The nullity is 0, and the rank is 2. Thus T is one-to-one, but not onto. 6. The nullity is n − 1, and the rank is 1. Thus T is not one-to-one unless n = 1, and T is not onto unless n = 1. 11. T (8 , 11) = (5 , − 3 , 16) 18. T ( a, b ) = ( b, 0). N ( T ) = span { (1 , 0) } = R ( T ). 19. Let V = W = R , and define T = I and U = 2 I . 23. All of R 3 or a plane in R 3 through the origin 25. (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) Let V = P ( F ) and W = span( { 1 } ). Define T first on the standard basis of V by T (1) = T ( x ) = 0 , and T ( x k ) = x k − 1 for k ≥ 2. Now extend T to a linear transformation from V to V . Then N ( T ) = span( { 1 , x } ), and R ( T ) = span( { x k : k ≥ 1 } ). So V = R ( T ) ⊕ W , but W 6 = N ( T ). 2.2 THE MATRIX REPRESENTATION OF A LINEAR TRANSFORMATION 2. (b) ( 2 3 − 1 1 0 1 ) (e)      1 0 . . . 0 1 0 . . . 0 . . . . . . . . . 1 0 . . . 0      4.   1 1 0 0 0 0 0 2 0 1 0 0   5. (c) (1 0 0 1) (d) (1 2 4) (f )   3 − 6 1   (g) (a) 4

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