Linear algebra and geometry: A second course

Irving Kaplansky9780828402798, 0828402795

Table of contents :
Title Page……Page 1
Preface……Page 5
A Note on Prerequisites……Page 7
Contents……Page 9
1-1 Definitions & Examples……Page 12
1-2 The Direct Summand Theorem……Page 16
1-3 Diagonalization……Page 18
1-4 The Inertia Theorem……Page 21
1-5 The Discriminant……Page 23
1-6 Finite Fields……Page 25
1-7 Witt’s Cancellation Theorem……Page 28
1-8 Hyperbolic Planes……Page 30
1-9 Alternate Forms……Page 32
1-10 Characteristic 2 : Symmetric Bilinear Forms……Page 34
1-11 Witt’s Theorem on Piecewise Equivalence……Page 36
1-12 Characteristic 2 : Quadratic Forms……Page 38
1-13 Hermitian Forms……Page 45
1-14 Some Alternative Proofs……Page 49
1-15 Infinite-Dimensional Inner Product Spaces……Page 50
1-16 Forms Over Rings……Page 57
2-1 The Real Self-Adioint Case……Page 68
2-2 Unitary Spaces……Page 71
2-3 Positivity & Polar Decomposition……Page 76
2-4 The Real Case, Continued……Page 80
2-5 Specht’s Theorem……Page 82
2-6 Remarks on Similarity……Page 84
2-7 Orthogonal Similarity over Algebraically Closed Fields……Page 88
3-1 Affine Planes……Page 94
3-2 Inner Product Planes……Page 104
3-3 Projective Planes……Page 108
3-4 Projective Transformations……Page 112
3-5 Duality……Page 115
3-6 Cross Ratio & Harmonic Range……Page 116
3-7 Conics……Page 124
3-8 Higher Dimensional Spaces……Page 131
3-9 Noncommutativity……Page 134
3-10 Synthetic Foundations of Geometry……Page 135
Appendix : Topological Aspects of Proiective Spaces……Page 141
Bibliography……Page 148
Index……Page 152