Linear algebra and geometry 线性代数与几何

A prominent and influential mathematician who has received numerous awards wrote this text to remedy a common failing in teaching algebra: the neglect of related instruction in geometry. Based on his many years of experience as an instructor the University of Chicago, author Irving Kaplansky presents a coherent overview of the correlation between these two branches of mathematics, illustrating his topics with an abundance of examples, exercises, and proofs. Suitable for both undergraduate and graduate courses. Unabridged republication of the edition published by Chelsea Publishing Company, New York, 1974.

1 INNER PRODUCT SPACES
1-1 Definitions and Examples
1-2 The Direct Summand Theorem
1-3 Diagonalization
1-4 The Inertia Theorem
1-5 The Discriminant
1-6 Finite Fields
1-7 Witt's Cancellation Theorem
1-8 Hyperbolic Planes
1-9 Alternate Forms
1-10 Characteristic 2: Symmetric Bilinear Forms
1-11 Witt's Theorem on Piecewise Equivalence
1-12 Characteristic 2: Quadratic Forms
1-13 Hermitian Forms
1-14 Some Alternative Proofs
1-15 Infinite-Dimensional Inner Product Spaces
1-16 Forms Over Rings
2　ORTHOGONAL SIMILARITY
2-1 The Real Self-Adjoint Case
2-2 Unitary Spaces
2-3 Positivity and Polar Decomposition
2-4 The Real Case, Continued
2-5 Specht's Theorem
2-6 Remarks on Similarity
2-7 Orthogonal Similarity over Algebraically Closed Fields
3 GEOMETRY
3-1 AfFine Planes
3-2 Inner Product Planes
3-3 Projective Planes
3-4 Projective Transformations
3-5 Duality
3-6 Cross Ratio and Harmonic Range
3-7 Conics
3-8 Higher Dimensional Spaces
3-9 Noncommutativity
3-10 Synthetic Foundations of Geometry
Appendix: Topological Aspects of Projective Spaces
Bibliography
Index