Part I

1. Vector Spaces

2. Linear Mappings

3. Structure of Vector Spaces

4. Matrices

5. Inner Product Spaces

6. Determinants (2 x 2 and 3 x 3)

Part II

7. Determinants (n x n)

8. Similarity (Act 1)

9. Euclidean Spaces (Spectral Theory)

10. Equivalence of Matrices over a PIR

11. Similarity (Act II)

12. Unitary Spaces

13. Tensor Products

Appendix A Foundations

Appendix B Integral Domains, Factorization Theory

Appendix C Weierstrass-Bolzano Theorem

Index of Notations

Index

Errata and Comments

A thorough first course in linear algebra, this two-part treatment begins with the basic theory of vector spaces and linear maps, including dimension, determinants, eigenvalues, and eigenvectors. The second section addresses more advanced topics such as the study of canonical forms for matrices.
The treatment can be tailored to satisfy the requirements of both introductory and advanced courses. Introductory courses that also serve as an initiation into formal mathematics will focus on the first six chapters. Students already schooled in matrices and linear mappings as well as theorem-proving will quickly proceed to selected chapters from part two. The selection can emphasize algebra or analysis/geometry, as needed. Ample examples, applications, and exercises appear throughout the text, which is supplemented by three helpful Appendixes.

Sterling K. Berberian is Professor Emeritus of Mathematics at the University of Texas at Austin and is the author of several books, including Introduction to Hilbert Space, Fundamentals of Real Analysis, and Measure and Integration.