Linear_Algebra-Friedburg-4e-solutions.pdf

Preface

This Instructor's Solutions Manual contains solutions for essentially all of the exercises in the

text that are intended to be done by hand. Solutions to Matlab exercises are not included.

The Student's Solutions Manual that accompanies this text contains solutions for only selected

odd-numbered exercises, including those exercises whose answers appear in the answer key. The

solutions that appear in the students' manual are identical to those provided in this manual,

and generally provide a more detailed solution than is available in the answer key. Although no

pattern is strictly adhered to throughout the student manual, the solutions provided there are

primarily to the computational exercises, whereas solutions that involve proof are generally not

included. None of the solutions to the supplementary end-of-chapter exercises are included in the

student manual.

Contents

Preface

iii

1 Matrices and Systems of Equations 1

1.1 Introduction to Matrices and Systems of Linear Equations . . . . . . . . . . . . . . 1

1.2 Echelon Form and Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . 6

1.3 Consistent Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Algebraic Properties of Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 Linear Independence and Nonsing. Matrices . . . . . . . . . . . . . . . . . . . . . . 26

1.8 Data ¯tting, Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.9 Matrix Inverses and their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.10 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.11 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Vectors in 2-Space and 3-Space 43

2.1 Vectors in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3 The Dot Product and the Cross Product . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Lines and Planes in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.6 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Vector Space Properties of R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Examples of Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Bases for Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.6 Orthogonal Bases for Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.7 Linear Transformations from R n

to R m . . . . . . . . . . . . . . . . . . . . . . . . 83

3 The Vector Space R n

vi CONTENTS

3.8 Least-Squares Solutions to Inconsistent Systems . . . . . . . . . . . . . . . . . . . . 89

3.9 Fitting Data and Least Squares Solutions . . . . . . . . . . . . . . . . . . . . . . . 92

3.10 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.11 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4 The Eigenvalue Problems 99

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Determinants and the Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . 101

4.3 Elementary Operations and Determinants . . . . . . . . . . . . . . . . . . . . . . . 104

4.4 Eigenvalues and the Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . 108

4.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.6 Complex Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.7 Similarity Transformations & Diagonalization . . . . . . . . . . . . . . . . . . . . . 121

4.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.9 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.10 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5 Vector Spaces and Linear Transformations 135

5.1 Introduction (No exercises) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.4 Linear Independence, Bases, and Coordinates . . . . . . . . . . . . . . . . . . . . . 144

5.5 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.6 Inner-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.7 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.8 Operations with Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . 158

5.9 Matrix Representations for Linear Transformations . . . . . . . . . . . . . . . . . . 161

5.10 Change of Basis and Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.11 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.12 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6 Determinants 175

6.1 Introduction (No exercises) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.2 Cofactor Expansion of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.3 Elementary Operations and Determinants . . . . . . . . . . . . . . . . . . . . . . . 178

6.4 Cramer's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.5 Applications of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.6 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.7 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7 Eigenvalues and Applications 193

7.1 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.2 Systems of Di®erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.3 Transformation to Hessenberg Form . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.4 Eigenvalues of Hessenberg Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.5 Householder Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

7.6 QR Factorization & Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.7 Matrix Polynomials & The Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . 211

7.8 Generalized Eigenvectors & Di®. Eqns. . . . . . . . . . . . . . . . . . . . . . . . . . 212

7.9 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.10 Conceptual Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Chapter 1

Matrices and Systems of Equations

1.1 Introduction to Matrices and Systems of Linear Equations

1. Linear.

2. Nonlinear.

3. Linear.

4. Nonlinear.

5. Nonlinear.

6. Linear.

7. x 1 +3x 2 = 7

4x 1 ¡ x 2 = 2

1+3¢2 = 7

4¢1¡2 = 2

8. 6x 1 ¡ x 2 + x 3 = 14

x 1 +2x 2 +4x 3 = 4

6¢2¡(¡1)+1 = 14

2+2¢(¡1)+4¢1 = 4

9. x 1 + x 2 = 0

3x 1 +4x 2 = ¡1

¡x 1 +2x 2 = ¡3

1+(¡1) = 0

3¢1+4¢(¡1) = ¡1

¡1+2¢(¡1) = ¡3

10. 3x 2 = 9;

4x 1 = 8;

3¢3 = 9

4¢2 = 8

11. Unique solution.

12. No Solution

13. In¯nitely many solutions.

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