Solutions for Linear Algebra, 5th Edition Friedberg (All Chapters included)

INSTRUCTOR’S
SOLUTIONS MANUAL

LINEAR A LGEBRA
FIFTH EDITION


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

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




** Immediate Download
** Swift Response
** All Chapters included

,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


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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




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