Linear Algebra 5th Edition by Stephen Friedberg, ISBN-13: 978-0134860244
[PDF eBook eTextbook] – Available Instantly
- 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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