A First Course in Linear Algebra

Title
Preface
Contents
Part One: Topics From Vector Geometry
Untitled
1. Vector Algebra in the Euclidean Plane
2. Vector Algebra in Euclidean Space
3. Parametric Equations of Lines in R^2 and R^3
4. The Locus of a Linear Equation in R^2 and R^3
5. Area of a Parallelogram and Volume of a Parallelepiped
6. Generalizations to R^n
Part Two: Matrices and Linear Equations
7. Matrices and Their Algebra
8. Systems of Linear Equations
9. lnvertible Matrices
10. The Jacobian Matrix of a Differentiable Map
Part Three: Optional Topics in Abstract Algebra
11. Semigroups
12. Groups
13. Multiplicative Notation Versus Additive Notation
14. Forming New Groups from Given Ones
15. Homomorphisms of Groups
16. The Group Sn
17. Rings
18. Fields
19. Homomorphisms of Rings
20. Polynomial Rings Over a Field
21. Vector Spaces Over Arbitrary Fields
Part Four: Fundamentals of Linear Algebra
22. Real Vector Spaces
23. Subspaces and Linear Combinations
24. Independent Sets
25. Bases and Dimension
26. Sums of Subspaces; Flats
27. Coordinatization of Vectors
28. Linear Maps
29. Coordinatization of a Linear Map
30. The Algebra of Linear Maps
31. Change of Basis
32. Local Approximation of Differentiable Maps
33. Rank
34. The General Linear Problem
35. Characteristic Values and Diagonalization
36. Determinants
37. Theory of Determinants
38. Two Classical Applications of Determinants
39. Characteristic Values and Determinants
Part Five: Additional Topics in Linear Algebra
40. The Jordan Canonical Form and Invariant Subspaces
41. Inner Products in Real Vector Spaces
42. Normed Vector Spaces
43. Orthogonal Bases
44. Unitary Maps and Matrices
45. The n-Volume of an n-Box in R^m
46. Linear Maps and Volumes
Appendix: Sets, Maps, and Relations
Answers to Odd-Numbered Exercises
Index