**Linear Algebra - As an Introduction to Abstract Mathematics**
is an introductory textbook designed for undergraduate mathematics majors
with an emphasis on abstraction and in particular the concept of
proofs in the setting of linear algebra. Typically such a student
will have taken calculus, though the only prerequisite is suitable
mathematical maturity. The purpose of this book is to bridge the gap
between the more conceptual and computational oriented lower division
undergraduate classes to the more abstract oriented upper division
classes. The book begins with systems of linear equations and complex
numbers, then relates these to the abstract notion of linear maps on
finite-dimensional vector spaces, and covers diagonalization,
eigenspaces, determinants, and the Spectral Theorem. Each chapter
concludes with both proof-writing and computational exercises.

**Content:**

1. What is linear algebra

2. Introduction to complex numbers

3. The fundamental theorem of algebra and factoring polynomials

4. Vector spaces

5. Span and bases

6. Linear maps

7. Eigenvalues and eigenvectors

8. Permutations and the determinant

9. Inner product spaces

10. Change of bases

11. The spectral theorem for normal linear maps

12. Supplementary notes on matrices and linear systems

Appendix: The language of sets and functions; algebraic structures encountered; common math symbols; notation used

This book has already been successfully used for the courses

- MAT67 Modern Linear Algebra at UC Davis
- MATH2730 Linear Algebra at Simpson University
- Math 413/513 Linear Algebra at Arizona University
- Lineare Algebra at Universitaet Leipzig