Geometric linear algebra

I-Hsiung Lin, Yixiong Lin9812569332, 9812560874, 9789812569332, 9789812560872, 9789812561329, 9812561323, 9789812707758, 9812707751

This accessible book for beginners uses intuitive geometric concepts to create abstract algebraic theory with a special emphasis on geometric characterizations. The book applies known results to describe various geometries and their invariants, and presents problems concerned with linear algebra, such as in real and complex analysis, differential equations, differentiable manifolds, differential geometry, Markov chains and transformation groups. The clear and inductive approach makes this book unique among existing books on linear algebra both in presentation and in content.

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Table of contents :
Cover……Page 1
Short title……Page 2
Title page……Page 3
Date-line……Page 4
PREFACE……Page 5
CONTENTS……Page 19
Part 1: The Affine and Linear Structures of $mathbb{R}^1$, $mathbb{R}^2$ and $mathbb{R}^3$……Page 22
Introduction……Page 26
1.1 Vectorization of a Straight Line: Affine Structure……Page 28
1.2 Coordinatization of a Straight Line: $mathbb{R}^1$ (or $mathbb{R}$)……Page 31
1.3 Changes of Coordinates: Affine and Linear Transformations (or Mappings)……Page 35
1.4 Affine Invariants……Page 38
Introduction……Page 42
Sketch of the Content……Page 44
2.1 (Plane) Vector……Page 45
2.2 Vectorization of a Plane: Affine Structure……Page 51
2.3 Coordinatization of a Plane: $mathbb{R}^2$……Page 55
2.4 Changes of Coordinates: Affine and Linear Transformations (or Mappings)……Page 66
2.5 Straight Lines in a Plane……Page 80
2.6 Affine and Barycentric Coordinates……Page 91
2.7 Linear Transformations (Operators)……Page 102
2.7.1 Linear operators in the Cartesian coordinate system……Page 107
2.7.2 Examples……Page 112
2.7.3 Matrix representations of a linear operator in various bases……Page 135
2.2.4 Linear transformations (operators)……Page 156
2.7.5 Elementary matrices and matrix factorizations……Page 169
2.7.6 Diagonal canonical form……Page 207
2.2.7 Jordan canonical form……Page 239
2.7.8 Rational canonical form……Page 251
2.8 Affine Transformations……Page 256
2.8.1 Matrix representations……Page 260
2.8.2 Examples……Page 271
2.8.3 Affine invariants……Page 306
2.8.4 Affine geometry……Page 313
2.8.5 Quadratic curves……Page 321
Introduction……Page 340
Sketch of the Content……Page 342
3.1 Vectorization of a Space: Affine Structure……Page 343
3.2 Coordinatization of a Space: $mathbb{R}^3$……Page 347
3.3 Changes of Coordinates: Affine Transformation (or Mapping)……Page 356
3.4 Lines in Space……Page 366
3.5 Planes in Space……Page 371
3.6 Affine and Barycentric Coordinates……Page 382
3.7.1 Linear operators in the Cartesian coordinate system……Page 386
3.7.2 Examples……Page 405
3.7.3 Matrix representations of a linear operator in various bases……Page 427
3.7.4 Linear transformations (operators)……Page 456
3.7.5 Elementary matrices and matrix factorizations……Page 463
3.7.6 Diagonal canonical form……Page 497
3.7.7 Jordan canonical form……Page 532
3.7.8 Rational canonical form……Page 579
3.8 Affine Transformations……Page 599
3.8.1 Matrix representations……Page 600
3.8.2 Examples……Page 611
3.8.3 Affine invariants……Page 657
3.8.4 Affine geometry……Page 661
3.8.5 Quadrics……Page 689
A.1 Sets……Page 702
A.2 Functions……Page 703
A.3 Fields……Page 705
A.4 Groups……Page 707
A.5 Polynomials……Page 708
B.1 Vector (or Linear) Spaces……Page 712
B.2 Main Techniques: Linear Combination, Dependence and Independence……Page 716
B.3 Basis and Dimension……Page 718
B.4 Matrices……Page 720
B.5 Elementary Matrix Operations and Row-Reduced Echelon Matrices……Page 740
B.6 Determinants……Page 748
B.7 Linear Transformations and Their Matrix Representations……Page 753
B.8 A Matrix and its Transpose……Page 777
B.9 Inner Product Spaces……Page 794
B.10 Eigenvalues and Eigenvectors……Page 811
B.11 Diagonalizability of a Square Matrix or a Linear Operator……Page 814
B.12.1 Jordan canonical form……Page 820
B.12.2 Rational canonical form……Page 830
References……Page 840
Index of Notations……Page 844
Index……Page 860