This is a pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. The reader should be already exposed to the array-based formalism of vector and matrix calculations. This book makes extensive use of the exterior (anti-commutative, "wedge") product of vectors. The coordinate-free formalism and the exterior product, while somewhat more abstract, provide a deeper understanding of the classical results in linear algebra. Without cumbersome matrix calculations, this text derives the standard properties of determinants, the Pythagorean formula for multidimensional volumes, the formulas of Jacobi and Liouville, the Cayley-Hamilton theorem, the Jordan canonical form, the properties of Pfaffians, as well as some generalizations of these results.
Author(s): Sergei Winitzki
Publisher: lulu.com
Year: 2010
Language: English
Pages: 299
Preface......Page 9
0.1 Notation......Page 11
0.2 Sample quiz problems......Page 13
0.3 A list of results......Page 15
1.1.1 Three-dimensional Euclidean geometry......Page 21
1.1.2 From three-dimensional vectors to abstract vectors......Page 22
1.1.3 Examples of vector spaces......Page 26
1.1.4 Dimensionality and bases......Page 29
1.1.5 All bases have equally many vectors......Page 34
1.2 Linear maps in vector spaces......Page 36
1.2.1 Abstract definition of linear maps......Page 37
1.2.2 Examples of linear maps......Page 39
1.2.4 Eigenvectors and eigenvalues......Page 41
1.3 Subspaces......Page 43
1.3.2 Eigenspaces......Page 44
1.4 Isomorphisms of vector spaces......Page 45
1.5 Direct sum of vector spaces......Page 46
1.6 Dual (conjugate) vector space......Page 47
1.6.1 Dual basis......Page 48
1.6.2 Hyperplanes......Page 53
1.7 Tensor product of vector spaces......Page 55
1.7.1 First examples......Page 57
1.7.2 Example: RmRn......Page 58
1.7.3 Dimension of tensor product is the product of dimensions......Page 59
1.7.5 * Distributivity of tensor product......Page 62
1.8.1 Tensors as linear operators......Page 63
1.8.2 Linear operators as tensors......Page 65
1.8.3 Examples and exercises......Page 67
1.8.4 Linear maps between different spaces......Page 70
1.9.1 Definition of index notation......Page 75
1.9.2 Advantages and disadvantages of index notation......Page 78
1.10.1 Definition of Dirac notation......Page 79
1.10.2 Advantages and disadvantages of Dirac notation......Page 81
2.1.1 Two-dimensional oriented area......Page 83
2.1.2 Parallelograms in R3 and in Rn......Page 86
2.2.1 Definition of exterior product......Page 88
2.2.2 * Symmetric tensor product......Page 94
2.3 Properties of spaces kV......Page 95
2.3.1 Linear maps between spaces kV......Page 97
2.3.2 Exterior product and linear dependence......Page 99
2.3.3 Computing the dual basis......Page 103
2.3.4 Gaussian elimination......Page 105
2.3.5 Rank of a set of vectors......Page 107
2.3.6 Exterior product in index notation......Page 109
2.3.7 * Exterior algebra (Grassmann algebra)......Page 113
3.1 Determinants through permutations: the hard way......Page 117
3.2 The space NV and oriented volume......Page 119
3.3 Determinants of operators......Page 124
3.3.1 Examples: computing determinants......Page 127
3.4 Determinants of square tables......Page 130
3.4.1 * Index notation for NV and determinants......Page 134
3.5 Solving linear equations......Page 136
3.5.1 Existence of solutions......Page 137
3.5.2 Kramer's rule and beyond......Page 139
3.6 Vandermonde matrix......Page 143
3.6.1 Linear independence of eigenvectors......Page 145
3.6.2 Polynomial interpolation......Page 146
3.7 Multilinear actions in exterior powers......Page 147
3.8 Trace......Page 151
3.9 Characteristic polynomial......Page 155
3.9.1 Nilpotent operators......Page 161
4.1.1 Exterior transposition of operators......Page 165
4.1.2 * Index notation......Page 168
4.2.1 Definition of algebraic complement......Page 170
4.2.2 Algebraic complement of a matrix......Page 174
4.2.3 Further properties and generalizations......Page 176
4.3 Cayley-Hamilton theorem and beyond......Page 180
4.4 Functions of operators......Page 183
4.4.1 Definitions. Formal power series......Page 184
4.4.2 Computations: Sylvester's method......Page 187
4.4.3 * Square roots of operators......Page 192
4.5 Formulas of Jacobi and Liouville......Page 197
4.5.1 Derivative of characteristic polynomial......Page 201
4.5.2 Derivative of a simple eigenvalue......Page 202
4.5.3 General trace relations......Page 204
4.6 Jordan canonical form......Page 205
4.6.1 Minimal polynomial......Page 211
4.7 * Construction of projectors onto Jordan cells......Page 214
5.1 Vector spaces with scalar product......Page 223
5.1.1 Orthonormal bases......Page 225
5.1.2 Correspondence between vectors and covectors......Page 229
5.1.3 * Example: bilinear forms on VV*......Page 231
5.2 Orthogonal subspaces......Page 232
5.2.1 Affine hyperplanes......Page 236
5.3.1 Examples and properties......Page 237
5.3.2 Transposition......Page 239
5.4.1 Orthonormal bases, volume, and NV......Page 240
5.4.2 Vector product in R3 and Levi-Civita symbol......Page 242
5.4.3 Hodge star and Levi-Civita symbol in N dimensions......Page 244
5.4.4 Reciprocal basis......Page 248
5.5.1 Scalar product in NV......Page 250
5.5.2 Volumes of k-dimensional parallelepipeds......Page 252
5.6 Scalar product for complex spaces......Page 256
5.6.1 Symmetric and Hermitian operators......Page 258
5.7 Antisymmetric operators......Page 261
5.8 * Pfaffians......Page 264
5.8.1 Determinants are Pfaffians squared......Page 266
5.8.2 Further properties......Page 268
A.1 Basic definitions......Page 271
A.2 Geometric representation......Page 272
A.4 Exponent and logarithm......Page 273
B Permutations......Page 275
C.1 Definitions......Page 279
C.2 Matrix multiplication......Page 280
C.3 Linear equations......Page 283
C.4 Inverse matrix......Page 284
C.5 Determinants......Page 285
C.6 Tensor product......Page 287
D.1 Motivation......Page 289
D.2.2 Applicability and definitions......Page 290
D.2.5 Modifications......Page 292
D.2.7 Collections of documents......Page 294
D.2.11 Future revisions of this license......Page 295
D.2.13 Copyright......Page 296
