**Linear Algebra 5th Edition by Stephen Friedberg, ISBN-13: 978-0134860244**

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- Publisher: Pearson; 5th edition (September 7, 2018)
- Language: English
- 608 pages
- ISBN-10: 0134860241
- ISBN-13: 978-0134860244

**For courses in Advanced Linear Algebra.**

Illustrates the power of linear algebra through practical applications.

This acclaimed theorem-proof text presents a careful treatment of the principal topics of linear algebra. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Applications to such areas as differential equations, economics, geometry, and physics appear throughout, and can be included at the instructor’s discretion.

**Table of Contents:**

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

An Application

Exercises

3.4 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

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.

6.6 Orthogonal Projections and the Spectral Theorem

Exercises

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

6.8* Bilinear and Quadratic Forms

Bilinear Forms

Definition.

Definitions.

Proof.

Definition.

Proof.

Corollary 1.

Proof.

Corollary 2.

Corollary 3.

Definition.

Proof.

Corollary.

Proof.

Symmetric Bilinear Forms

Definition.

Proof.

Definition.

Corollary.

Proof.

Proof.

Proof.

Corollary.

Proof.

Diagonalization of Symmetric Matrices

Quadratic Forms

Definition.

Quadratic Forms on a Real Inner Product Space

Proof.

Corollary.

Proof.

The Second Derivative Test for Functions of Several Variables

Proof.

Sylvester’s Law of Inertia

Proof.

Definitions.

Corollary 1 (Sylvester’s Law of Inertia for Matrices).

Definitions.

Corollary 2.

Proof.

Corollary 3.

Exercises

6.9* 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

* Stephen H. Friedberg *holds a BA in mathematics from Boston University and MS and PhD degrees in mathematics from Northwestern University, and was awarded a Moore Postdoctoral Instructorship at MIT. He served as a director for CUPM, the Mathematical Association of America’s Committee on the Undergraduate Program in Mathematics. He was a faculty member at Illinois State University for 32 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1990. He has also taught at the University of London, the University of Missouri and at Illinois Wesleyan University. He has authored or coauthored articles and books in analysis and linear algebra.

* Arnold J. Insel *received BA and MA degrees in mathematics from the University of Florida and a PhD from the University of California at Berkeley. He served as a faculty member at Illinois State University for 31 years and at Illinois Wesleyan University for 2 years. In addition to authoring and co-authoring articles and books in linear algebra, he has written articles in lattice theory, topology and topological groups.

* Lawrence E. Spence *holds a BA from Towson State College and MS and PhD degrees in mathematics from Michigan State University. He served as a faculty member at Illinois State University for 34 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1987. He is an author or co-author of 9 college mathematics textbooks, as well as articles in mathematics journals in the areas of discrete mathematics and linear algebra.

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