Linear Algebra as an Introduction to Abstract Mathematics

This 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 would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented 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.

Contents:

• What is Linear Algebra

• Introduction to Complex Numbers

• The Fundamental Theorem of Algebra and Factoring Polynomials

• Vector Spaces

• Span and Bases

• Linear Maps

• Eigenvalues and Eigenvectors

• Permutations and the Determinant of a Square Matrix

• Inner Product Spaces

• Change of Bases

• The Spectral Theorem for Normal Linear Maps

• Appendices:

  • Supplementary Notes on Matrices and Linear Systems
  • The Language of Sets and Functions
  • Summary of Algebraic Structures Encountered
  • Some Common Math Symbols and Abbreviations

Readership: Undergraduates in mathematics.