Linear Algebra

About our authors 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.

* Sections denoted by an asterisk are optional.



Vector Spaces

1.1 Introduction
1.2 Vector Spaces
1.3 Subspaces
1.4 Linear Combinations and Systems of Linear Equations
1.5 Linear Dependence and Linear Independence
1.6 Bases and Dimension
1.7* Maximal Linearly Independent Subsets



Index of Definitions


Linear Transformations and Matrices

2.1 Linear Transformations, Null Spaces, and Ranges
2.2 The Matrix Representation of a Linear Transformation
2.3 Composition of Linear Transformations and Matrix Multiplication
2.4 Invertibility and Isomorphisms
2.5 The Change of Coordinate Matrix
2.6* Dual Spaces
2.7* Homogeneous Linear Differential Equations with Constant Coefficients



Index of Definitions


Elementary Matrix Operations and Systems of Linear Equations

3.1 Elementary Matrix Operations and Elementary Matrices
3.2 The Rank of a Matrix and Matrix Inverses
3.3 Systems of Linear Equations – Theoretical Aspects
3.4 Systems of Linear Equations – Computational Aspects



Index of Definitions


Determinants

4.1 Determinants of Order 2
4.2 Determinants of Order n
4.3 Properties of Determinants
4.4 Summary|Important Facts about Determinants
4.5* A Characterization of the Determinant



Index of Definitions


Diagonalization

5.1 Eigenvalues and Eigenvectors
5.2 Diagonalizability
5.3* Matrix Limits and Markov Chains
5.4 Invariant Subspaces and the Cayley–Hamilton Theorem



Index of Definitions


Inner Product Spaces

6.1 Inner Products and Norms
6.2 The Gram–Schmidt Orthogonalization Process and Orthogonal Complements
6.3 The Adjoint of a Linear Operator
6.4 Normal and Self-Adjoint Operators
6.5 Unitary and Orthogonal Operators and Their Matrices
6.6 Orthogonal Projections and the Spectral Theorem
6.7* The Singular Value Decomposition and the Pseudoinverse
6.8* Bilinear and Quadratic Forms
6.9* Einstein's Special Theory of Relativity
6.10* Conditioning and the Rayleigh Quotient
6.11* The Geometry of Orthogonal Operators



Index of Definitions


Canonical Forms

7.1 The Jordan Canonical Form I
7.2 The Jordan Canonical Form II
7.3 The Minimal Polynomial
7.4* The Rational Canonical Form



Index of Definitions



Appendices

A. Sets
B. Functions
C. Fields
D. Complex Numbers
E. Polynomials

Answers to Selected Exercises

Index