Linear Algebra - As an Introduction to Abstract Mathematics

Free online text by Isaiah Lankham, Bruno Nachtergaele and Anne Schilling

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.

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

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