SOLUTIONS MANUAL - INSTRUCTOR’S Complete Chapter Solutions Manua...

SOLUTIONS MANUAL'

LINEAR ALGEBRA

FIFTH EDITION

Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Illinois State University

INSTRUCTOR’S

Complete Chapter Solutions Manual are included (Ch 1 to 7) *

• Immediate Download

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• Swift Response

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• All Chapters included 1 / 3

Contents

• Vector Spaces1

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

• Linear Transformations and Matrices4

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 Dierential Equations with Constant Coecients . . . . . . . . 6

• 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

• Determinants11

4.1 Determinants of Order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Determinants of Ordern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.4 Summary{Important Facts about Determinants . . . . . . . . . . . . . . . . . . . . 12 4.5 A Characterization of the Determinant . . . . . . . . . . . . . . . . . . . . . . . . . 12

• Diagonalization13

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 2 / 3

Table of Contents

• Inner Product Spaces16

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

• Canonical Forms21

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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