Linear Algebra and Geometry: Second Course

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