Linear algebra is now included in the undergraduate curriculum of most universities. It is generally recognized that this branch of algebra, being less abstract and directly motivated by geometry, is easier to understand than some other branches and that because of the wide applications it should be taught as soon as possible. This book is an extension of the lecture notes for a course in algebra and geometry for first-year undergraduates of mathematics and physical sciences.
Author(s): Kam-Tim Leung
Publisher: Hong Kong University Press
Year: 1974
Language: English
Pages: 318
Tags: Математика;Линейная алгебра и аналитическая геометрия;Линейная алгебра;
Cover......Page 1
The Author ......Page 2
Title Page ......Page 4
verso ......Page 5
Preface ......Page 6
Table of Contents ......Page 8
Ch. I. LINEAR SPACE ......Page 12
A. Abelian groups ......Page 15
B. Linear spaces ......Page 21
C. Examples ......Page 22
D. Exercises ......Page 26
A. Linear combinations ......Page 28
B. Base ......Page 30
C. Linear independence ......Page 31
D. Dimension ......Page 34
E. Coordinates ......Page 38
F. Exercises ......Page 40
A. Existence of Base ......Page 43
B. Dimension ......Page 44
C. Exercises ......Page 45
A. General properties ......Page 46
B. Operations on subspaces ......Page 47
C. Direct sum ......Page 49
D. Quotient space ......Page 51
E. Exercises ......Page 52
A. Linear transformation and examples ......Page 56
B. ......Page 61
C. Isomorphism ......Page 63
D. Kernel and image ......Page 65
E. Factorization ......Page 67
F. Exercises ......Page 69
A. The algebraic structure of Hom(X, Y) ......Page 73
B. The associative algebra End(X) ......Page 77
C. Direct sum and direct product ......Page 78
D. Exercises ......Page 81
7. Dual Space ......Page 84
A. General properties of dual space ......Page 85
B. Dual transformations ......Page 87
C. Natural transformations ......Page 88
D. A duality between L(X) and L(X*) ......Page 91
E. Exercises ......Page 93
8. The Category of Linear Spaces ......Page 95
A. Category ......Page 96
B. Functor ......Page 100
C. Natural transformation ......Page 102
D. Exercises ......Page 104
9. Affine Space ......Page 107
A. Points and vectors ......Page 109
B. Barycentre ......Page 112
C. Linear varieties ......Page 115
D. Lines ......Page 118
E. Base ......Page 119
F. Exercises ......Page 122
A. General properties ......Page 124
B. The category of affine spaces ......Page 127
A. Points at infinity ......Page 129
B. Definition of projective space ......Page 133
C. Homogeneous coordinates ......Page 135
D. Linear variety ......Page 136
E. The theorems of Pappus and Desargues ......Page 137
F. Cross ratio ......Page 141
G. Linear construction ......Page 144
H. The principle of duality ......Page 148
I. Exercises ......Page 150
12. Mappings of Projective Spaces ......Page 152
A. Projective isomorphism ......Page 153
B. Projectivities ......Page 154
C. Semi-linear transformations ......Page 157
D. The projective group ......Page 163
E. Exercises ......Page 164
A. Notations ......Page 166
B. Addition and scalar multiplication of matrices ......Page 170
C. Product of matrices ......Page 171
D. Exercises ......Page 174
A. Matrix of a linear transformation ......Page 177
B. Square matrices ......Page 179
C. Change of bases ......Page 181
D. Exercises ......Page 182
15. Systems of Linear Equations ......Page 186
A. The rank of a matrix ......Page 187
B. The solutions of a system of linear equations ......Page 188
C. Elementary transformations on matrices ......Page 189
D. Parametric representation of solutions ......Page 194
E. Two interpretations of elementary transformations on matrices ......Page 197
F. Exercises ......Page 202
Ch. VI. MULTILINEAR FORMS ......Page 207
A. Bilinear mappings ......Page 208
B. Quadratic forms ......Page 212
C. Multilinear forms ......Page 213
D. Exercises ......Page 215
17. Determinants ......Page 217
A. Determinants of order 3 ......Page 218
B. Permutations ......Page 220
C. Determinant functions ......Page 223
D. Determinants ......Page 225
E. Some useful rules ......Page 229
F. Cofactors and minors ......Page 231
G. Exercises ......Page 236
A. Definitions ......Page 241
B. Euclidean algorithm ......Page 243
C. Greatest common divisor ......Page 245
D. Substitutions ......Page 246
E. Exercises ......Page 248
A. Invariant subspaces ......Page 250
B. Eigenvectors and eigenvalues ......Page 253
C. Characteristic polynomials ......Page 254
D. Diagonalizable endomorphisms ......Page 256
E. Exercises ......Page 259
20. Jordan Form ......Page 261
A. Triangular form ......Page 262
B. Hamilton-Cayley theorem ......Page 264
C. Canonical decomposition ......Page 266
D. Nilpotent endomorphisms ......Page 270
E. Jordan theorem ......Page 273
F. Exercises ......Page 276
Ch. VIII. INNER PRODUCT SPACES ......Page 278
A. Inner product and norm ......Page 279
B. Orthogonality ......Page 282
C. SCHWARZ'S inequality ......Page 286
E. Exercises ......Page 289
A. The conjugate isomorphism ......Page 291
B. The adjoint transformation ......Page 293
C. Self-adjoint linear transformations ......Page 296
D. Eigenvalues of self-adjoint transformations ......Page 297
E. Bilinear forms on a euclideanspace ......Page 300
F. Isometry ......Page 302
G. Exercises ......Page 305
A. Orthogonality ......Page 308
C. The adjoint ......Page 311
D. Self-adjoint transformations ......Page 312
F. Normal transformation ......Page 313
G. Exercises ......Page 315
Index ......Page 317
