1 Matrix Algebra ............................................... 1
1.1 Vectors, Matrices, and Basic Operations on Them ......... 3
1.1.1 Problems for Practice ........................... 10
1.2 Matrices as Linear Maps ................................ 11
1.2.1 Problems for Practice ........................... 14
1.3 Matrix Multiplication .................................. 15
1.3.1 Problems for Practice ........................... 26
1.4 Block Matrices ......................................... 29
1.4.1 Problems for Practice ........................... 31
1.5 Invertible Matrices .................................... 31
1.5.1 Problems for Practice ........................... 41
1.6 The Transpose of a Matrix .............................. 44
1.6.1 Problems for Practice ........................... 51
2 Square Matrices of Order 2 .................................. 53
2.1 The Trace and the Determinant Maps ..................... 53
2.1.1 Problems for Practice ........................... 56
2.2 The Characteristic Polynomial and the Cayley-Hamilton
Theorem ................................................ 57
2.2.1 Problems for Practice ........................... 65
2.3 The Powers of a Square Matrix of Order 2 ............... 67
2.3.1 Problems for Practice ........................... 70
2.4 Application to Linear Recurrences ...................... 70
2.4.1 Problems for Practice ........................... 73
2.5 Solving the Equation Xn = A ............................ 74
2.5.1 Problems for Practice ........................... 78
2.6 Application to Pell's Equations ........................ 79
2.6.1 Problems for Practice ........................... 83
3 Matrices and Linear Equations ............................... 85
3.1 Linear Systems: The Basic Vocabulary ................... 85
3.1.1 Problems for Practice ........................... 87
3.2 The Reduced Row-Echelon form and Its Relevance to
Linear Systems ......................................... 88
3.2.1 Problems for Practice ........................... 95
3.3 Solving the System AX = b .............................. 96
3.3.1 Problems for Practice ........................... 99
3.4 Computing the Inverse of a Matrix ..................... 100
3.4.1 Problems for Practice .......................... 105
4 Vector Spaces and Subspaces ................................ 107
4.1 Vector Spaces-Definition, Basic Properties and
Examples .............................................. 107
4.1.1 Problems for Practice .......................... 113
4.2 Subspaces ............................................. 114
4.2.1 Problems for Practice .......................... 121
4.3 Linear Combinations and Span .......................... 122
4.3.1 Problems for Practice .......................... 127
4.4 Linear Independence ................................... 128
4.4.1 Problems for Practice .......................... 133
4.5 Dimension Theory ...................................... 135
4.5.1 Problems for Practice .......................... 146
5 Linear Transformations ..................................... 149
5.1 Definitions and Objects Canonically Attached to
a Linear Map .......................................... 149
5.1.1 Problems for practice .......................... 157
5.2 Linear Maps and Linearly Independent Sets ............. 159
5.2.1 Problems for practice .......................... 163
5.3 Matrix Representation of Linear Transformations ....... 164
5.3.1 Problems for practice .......................... 181
5.4 Rank of a Linear Map and Rank of a Matrix ............. 183
5.4.1 Problems for practice .......................... 194
6 Duality .................................................... 197
6.1 The Dual Basis ........................................ 197
6.1.1 Problems for Practice .......................... 208
6.2 Orthogonality and Equations for Subspaces ............. 210
6.2.1 Problems for Practice .......................... 218
6.3 The Transpose of a Linear Transformation .............. 220
6.3.1 Problems for Practice .......................... 224
6.4 Application to the Classification of Nilpotent
Matrices .............................................. 225
6.4.1 Problems for Practice .......................... 234
7 Determinants ............................................... 237
7.1 Multilinear Maps ...................................... 238
7.1.1 Problems for Practice .......................... 242
7.2 Determinant of a Family of Vectors, of a Matrix,
and of a Linear Transformation ........................ 243
7.2.1 Problems for Practice .......................... 251
7.3 Main Properties of the Determinant of a Matrix ........ 253
7.3.1 Problems for Practice .......................... 262
7.4 Computing Determinants in Practice .................... 264
7.4.1 Problems for Practice .......................... 278
7.5 The Vandermonde Determinant ........................... 282
7.5.1 Problems for Practice .......................... 287
7.6 Linear Systems and Determinants ....................... 288
7.6.1 Problems for Practice .......................... 298
8 Polynomial Expressions of Linear Transformations and
Matrices ................................................... 301
8.1 Some Basic Constructions .............................. 301
8.1.1 Problems for Practice .......................... 303
8.2 The Minimal Polynomial of a Linear Transformation or
Matrix ................................................ 304
8.2.1 Problems for Practice .......................... 309
8.3 Eigenvectors and Eigenvalues .......................... 310
8.3.1 Problems for Practice .......................... 316
8.4 The Characteristic Polynomial ......................... 319
8.4.1 Problems for Practice .......................... 330
8.5 The Cayley-Hamilton Theorem ........................... 333
8.5.1 Problems for Practice .......................... 337
9 Diagonalizability .......................................... 339
9.1 Upper-Triangular Matrices, Once Again ................. 340
9.1.1 Problems for Practice .......................... 343
9.2 Diagonalizable Matrices and Linear Transformations .... 345
9.2.1 Problems for Practice .......................... 356
9.3 Some Applications of the Previous Ideas ............... 359
9.3.1 Problems for Practice .......................... 372
10 Forms ...................................................... 377
10.1 Bilinear and Quadratic Forms .......................... 378
10.1.1 Problems for Practice .......................... 389
10.2 Positivity, Inner Products, and the Cauchy-Schwarz
Inequality ............................................ 391
10.2.1 Practice Problems .............................. 397
10.3 Bilinear Forms and Matrices ........................... 399
10.3.1 Problems for Practice .......................... 406
10.4 Duality and Orthogonality ............................. 408
10.4.1 Problems for Practice .......................... 416
10.5 Orthogonal Bases ...................................... 418
10.5.1 Problems for Practice .......................... 436
10.6 The Adjoint of a Linear Transformation ................ 442
10.6.1 Problems for Practice .......................... 448
10.7 The Orthogonal Group .................................. 450
10.7.1 Problems for Practice .......................... 465
10.8 The Spectral Theorem for Symmetric Linear
Transformations and Matrices .......................... 469
10.8.1 Problems for Practice .......................... 477
11 Appendix: Algebraic Prerequisites .......................... 483
11.1 Groups ................................................ 483
11.2 Permutations .......................................... 484
11.2.1 The Symmetric Group Sn ......................... 484
11.2.2 Transpositions as Generators of Sn ............. 486
11.2.3 The Signature Homomorphism ..................... 487
11.3 Polynomials ........................................... 489
References .................................................... 491
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