This book is an introduction to linear algebra, designed as a textbook for upper-division courses. It includes the basic results on vector spaces over fields, determinants, the theory of single linear transformation, and inner product spaces. The text presents proofs of the main theorems and includes numerical examples and exercises; applications to geometry, group theory and differential equations are given.

1. Introduction to Linear Algebra.- 1. Some problems which lead to linear algebra.- 2. Number systems and mathematical induction.- 2. Vector Spaces and Systems of Linear Equations.- 3. Vector spaces.- 4. Subspaces and linear dependence.- 5. The concepts of basis and dimension.- 6. Row equivalence of matrices.- 7. Some general theorems about finitely generated vector spaces.- 8. Systems of linear equations.- 9. Systems of homogeneous equations.- 10. Linear manifolds.- 3. Linear Transformations and Matrices.- 11. Linear transformations.- 12. Addition and multiplication of matrices.- 13. Linear transformations and matrices.- 4. Vector Spaces with an Inner Product.- 14. The concept of symmetry.- 15. Inner products.- 5. Determinants.- 16. Definition of determinants.- 17. Existence and uniqueness of determinants.- 18. The multiplication theorem for determinants.- 19. Further properties of determinants.- 6. Polynomials and Complex Numbers.- 20. Polynomials.- 21. Complex numbers.- 7. The Theory of a Single Linear Transformation.- 22. Basic concepts.- 23. Invariant subspaces.- 24. The triangular form theorem.- 25. The rational and Jordan canonical forms.- 8. Dual Vector Spaces and Multilinear Algebra.- 26. Quotient spaces and dual vector spaces.- 27. Bilinear forms and duality.- 28. Direct sums and tensor products.- 29. A proof of the elementary divisor theorem.- 9. Orthogonal and Unitary Transformations.- 30. The structure of orthogonal transformations.- 31. The principal axis theorem.- 32. Unitary transformations and the spectral theorem.- 10. Some Applications of Linear Algebra.- 33. Finite symmetry groups in three dimensions.- 34. Application to differential equations.- 35. Analytic methods in matrix theory.- 36. Sums of squares and Hurwitz's theorem.- Bibliography (with Notes).- Solutions of Selected Exercises.- Symbols (Including Greek Letters).