Geometric Linear Algebra : Volume One

by Lin, I-Hsiung; Lin, Yixiong

• ISBN13: 9789812561329
• ISBN10: 9812561323
• Format: Paperback
• Copyright: 2005-03-21
• Publisher: World Scientific Pub Co Inc

Summary

- Contains over 250 figures and numerous examples and challenging exercisesbull; Provides intensive applications of eigenvalues to geometric problemsbull; Almost every algebraic (computational) process is guided by some geometric way of thinking or by the use of or graphics

Table of Contents

Preface	vii
Part 1: The Affine and Linear Structures of R¹, R² and R³	1	(680)
Chapter 1 The One-Dimensional Real Vector Space JR (or R¹)	5	(16)
Introduction	5	(2)
Sketch of the Content	7	(1)
1.1 Vectorization of a Straight Line: Affine Structure	7	(3)
1.2 Coordinatization of a Straight Line: R¹ (or R)	10	(4)
1.3 Changes of Coordinates: Affine and Linear Transformations (or Mappings)	14	(3)
1.4 Affine Invariants	17	(4)
Chapter 2 The Two-Dimensional Real Vector Space R²	21	(298)
Introduction	21	(2)
Sketch of the Content	23	(1)
2.1 (Plane) Vector	24	(6)
2.2 Vectorization of a Plane: Affine Structure	30	(4)
2.3 Coordinatization of a Plane: R²	34	(11)
2.4 Changes of Coordinates: Affine and Linear Transformations (or Mappings)	45	(14)
2.5 Straight Lines in a Plane	59	(11)
2.6 Affine and Barycentric Coordinates	70	(11)
2.7 Linear Transformations (Operators)	81	(154)
2.7.1 Linear operators in the Cartesian coordinate system	86	(5)
2.7.2 Examples	91	(23)
2.7.3 Matrix representations of a linear operator in various bases	114	(21)
2.2.4 Linear transformations (operators)	135	(13)
2.7.5 Elementary matrices and matrix factorizations	148	(38)
2.7.6 Diagonal canonical form	186	(32)
2.2.7 Jordan canonical form	218	(12)
2.7.8 Rational canonical form	230	(5)
2.8 Affine Transformations	235	(84)
2.8.1 Matrix representations	239	(11)
2.8.2 Examples	250	(35)
2.8.3 Affine invariants	285	(7)
2.8.4 Affine geometry	292	(8)
2.8.5 Quadratic curves	300	(19)
Chapter 3 The Three-Dimensional Real Vector Space R³	319	(362)
Introduction	319	(2)
Sketch of the Content	321	(1)
3.1 Vectorization of a Space: Affine Structure	322	(4)
3.2 Coordinatization of a Space: R³	326	(9)
3.3 Changes of Coordinates: Affine Transformation (or Mapping)	335	(10)
3.4 Lines in Space	345	(5)
3.5 Planes in Space	350	(11)
3.6 Affine and Barycentric Coordinates	361	(4)
3.7 Linear Transformations (Operators)	365	(213)
3.7.1 Linear operators in the Cartesian coordinate system	365	(19)
3.7.2 Examples	384	(22)
3.7.3 Matrix representations of a linear operator in various bases	406	(29)
3.7.4 Linear transformations (operators)	435	(7)
3.7.5 Elementary matrices and matrix factorizations	442	(34)
3.7.6 Diagonal canonical form	476	(35)
3.7.7 Jordan canonical form	511	(47)
3.7.8 Rational canonical form	558	(20)
3.8 Affine Transformations	578	(221)
3.8.1 Matrix representations	579	(11)
3.8.2 Examples	590	(46)
3.8.3 Affine invariants	636	(4)
3.8.4 Affine geometry	640	(28)
3.8.5 Quadrics	668	(13)
Appendix A Some Prerequisites	681	(10)
A.1 Sets	681	(1)
A.2 Functions	682	(2)
A.3 Fields	684	(2)
A.4 Groups	686	(1)
A.5 Polynomials	687	(4)
Appendix B Fundamentals of Algebraic Linear Algebra	691	(128)
B.1 Vector (or Linear) Spaces	691	(4)
B.2 Main Techniques: Linear Combination, Dependence and Independence	695	(2)
B.3 Basis and Dimension	697	(2)
B.4 Matrices	699	(20)
B.5 Elementary Matrix Operations and Row-Reduced Echelon Matrices	719	(8)
B.6 Determinants	727	(5)
B.7 Linear Transformations and Their Matrix Representations	732	(24)
B.8 A Matrix and its Transpose	756	(17)
B.9 Inner Product Spaces	773	(17)
B.10 Eigenvalues and Eigenvectors	790	(3)
B.11 Diagonalizability of a Square Matrix or a Linear Operator	793	(6)
B.12 Canonical Forms for Matrices: Jordan Form and Rational Form	799	(20)
B.12.1 Jordan canonical form	799	(10)
B.12.2 Rational canonical form	809	(10)
References	819	(4)
Index of Notations	823	(16)
Index	839