This is a leisurely and mostly traditional introductory text in linear algebra. It tries to find a balance between abstract and concrete by defining all the concepts abstractly (for example, vector spaces and inner products) and using abstract terminology (for example, spectral theorem), but motivating them from Euclidean spaces and applying them mostly to Euclidean spaces. Despite these abstract trappings, it is still mostly a matrices-and-determinants book. Solutions to most problems are given in the back, and there are appendices dealing with set theory and with mathematical induction. The present book is a 2011 reprint of the 1988 Harcourt Brace Jovanovich edition, with errata sheets bound in.
The book is aimed fairly low, and despite its thickness only covers the beginning of linear algebra. It is aimed at a one-semester course for sophomores, although there is enough material for a full year, and there are several ways of selecting material for a one-semester course. Conspicuously absent are any numerical methods and considerations. It does not cover any of the various decompositions (QR, LU, SVD), although it does have a good bit on Jordan canonical form. The discussion of eigenvectors is thorough and includes the spectral theorem for Euclidean spaces.
For applications the book draws on two aspects of differential equations: systems of linear equations with constant coefficients, which are investigated thoroughly and use diagonalization, and a brief look at eigenfunction expansions for differential equations.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.
|A Guide to the Exercises|
|1. Vector Spaces|
|2. Linear Transformations|
|3. The Determinant Function|
|4. Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in Rn|
|5. Complex Numbers and Complex Vector Spaces|
|6. Jordan Canonical Form|
|7. Differential Equations|
|Appendix 1: Some Basic Logic and Set Theory|
|Appendix 2: Mathematical Induction|