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Answers to Selected Exercises
by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. S
Linear Algebra, 5th Edition
Linear Algebra
Contents
Preface
Suggested Course Outlines
Overview of Contents
Differences Between the Fourth and Fifth Editions
To the Student
1 Vector Spaces
1.1 Introduction
Exercises
1.2 Vector Spaces
Exercises
1.3 Subspaces
Exercises
Definition.
Definition.
1.4 Linear Combinations and Systems of Linear Equations
Exercises
1.5 Linear Dependence and Linear Independence
Exercises
1.6 Bases and Dimension
An Overview of Dimension and Its Consequences
The Dimension of Subspaces
Corollary.
The Lagrange Interpolation Formula
Exercises
1.7* Maximal Linearly Independent Subsets
Exercises
Index of Definitions for Chapter 1
2 Linear Transformations and Matrices
2.1 Linear Transformations, Null Spaces, and Ranges
Exercises
Definitions.
Definitions.
2.2 The Matrix Representation of a Linear Transformation
Exercises
2.3 Composition of Linear Transformations and Matrix Multiplication
Applications*
Exercises
2.4 Invertibility and Isomorphisms
Exercises
2.5 The Change of Coordinate Matrix
Exercises
2.6* Dual Spaces
Exercises
2.7* Homogeneous Linear Differential Equations with Constant Coefficients
Exercises
Index of Definitions for Chapter 2
3 Elementary Matrix Operations and Systems of Linear Equations
3.1 Elementary Matrix Operations and Elementary Matrices
Exercises
3.2 The Rank of a Matrix and Matrix Inverses
The Inverse of a Matrix
Definition.
Exercises
3.3 Systems of Linear Equations—Theoretical Aspects Systems of Linear Equations—Theoretical Aspects
An Application
Exercises
3.4 Systems of Linear Equations—Computational Aspects Systems of Linear Equations—Computational Aspects
Exercises
Index of Definitions for Chapter 3
4 Determinants
4.1 Determinants of Order 2
Exercises
4.2 Determinants of Order n
Exercises
4.3 Properties of Determinants
Exercises
Definition.
4.4 Summary—Important Facts about Determinants Summary—Important Facts about Determinants
Properties of the Determinant
Exercises
4.5* A Characterization of the Determinant
Exercises
Index of Definitions for Chapter 4
5 Diagonalization
5.1 Eigenvalues and Eigenvectors
Exercises
5.2 Diagonalizability
Test for Diagonalizability
Systems of Differential Equations
Direct Sums*
Definition.
Definition.
Proof.
Proof.
Exercises
Definitions.
5.3* Matrix Limits and Markov Chains
Applications*
Exercises
Definition.
5.4 Invariant Subspaces and the Cayley-Hamilton Theorem
The Cayley-Hamilton Theorem
Proof.
Corollary (Cayley-Hamilton Theorem for Matrices).
Proof.
Invariant Subspaces and Direct Sums3
Proof.
Definition.
Proof.
Exercises
Definition.
Index of Definitions for Chapter 5
6 Inner Product Spaces
6.1 Inner Products and Norms
Exercises
Definition.
6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements
Exercises
6.3 The Adjoint of a Linear Operator
Proof.
Proof.
Proof.
Corollary.
Proof.
Proof.
Corollary.
Proof.
Least Squares Approximation
Lemma 1.
Proof.
Lemma 2.
Proof.
Corollary.
Minimal Solutions to Systems of Linear Equations
Proof.
Exercises
Definition.
6.4 Normal and Self-Adjoint Operators
Exercises
Definitions.
6.5 Unitary and Orthogonal Operators and Their Matrices
Rigid Motions*
Definition.
Proof.
Orthogonal Operators on R2
Proof.
Corollary.
Conic Sections
Exercises
Definition.
6.7* The Singular Value Decomposition and the Pseudoinverse
The Polar Decomposition of a Square Matrix
Proof.
The Pseudoinverse
Definition.
The Pseudoinverse of a Matrix
The Pseudoinverse and Systems of Linear Equations
Proof.
Proof.
Exercises
6.9* Einstein’s Special Theory of Relativity Einstein’s Special Theory of Relativity
Time Contraction
Exercises
6.10* Conditioning and the Rayleigh Quotient
Exercises
6.11* The Geometry of Orthogonal Operators
Exercises
Index of Definitions for Chapter 6
7 Canonical Forms
7.1 The Jordan Canonical Form I
Exercises
7.2 The Jordan Canonical Form II
Exercises
Definitions.
Definition.
7.3 The Minimal Polynomial
Exercises
Definition.
7.4* The Rational Canonical Form
Uniqueness of the Rational Canonical Form
Lemma 1.
Lemma 2.
Corollary.
Definitions.
Direct Sums*
Proof.
Proof.
Exercises
Index of Definitions for Chapter 7
Appendices
Appendix A Sets
Appendix B Functions
Appendix C Fields
Definitions.
Proof.
Corollary.
Proof.
Proof.
Corollary.
Appendix D Complex Numbers
Definitions.
Proof.
Definition.
Proof.
Definition.
Proof.
Proof.
Corollary.
Proof.
Appendix E Polynomials
Definition.
Proof.
Corollary 1.
Proof.
Corollary 2.
Proof.
Definition.
Lemma.
Proof.
Proof.
Definitions.
Proof.
Proof.
Proof.
Definitions.
Proof.
Proof.
Proof.
Corollary.
Proof.
Proof.
Proof.
Answers to Selected Exercises
Chapter 1
Section 1.1
Section 1.2
Section 1.3
Section 1.4
Section 1.5
Section 1.6
Section 1.7
Chapter 2
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 2.5
Section 2.6
Section 2.7
Chapter 3
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Chapter 4
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Section 4.5
Chapter 5
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Chapter 6
Section 6.1
Section 6.2
Section 6.3
Section 6.4
Section 6.5
Section 6.6
Section 6.7
Section 6.8
Section 6.9
Section 6.10
Section 6.11
Chapter 7
Section 7.1
Section 7.2
Section 7.3
Section 7.4
Index
List of Symbols
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Appendix E Polynomials
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Chapter 1
Answers to Selected Exercises
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