Linear Algebra

Contents......Page 6
1.1 Introduction......Page 8
1.2 Vectors in R[sup(n)]......Page 9
1.3 Vector Addition and Scalar Multiplication......Page 10
1.4 Dot (Inner) Product......Page 11
1.5 Located Vectors, Hyperplanes, Lines, Curves in R[sup(n)]......Page 14
1.6 Vectors in R[sup(3)] (Spatial Vectors), ijk Notation......Page 16
1.7 Complex Numbers......Page 18
1.8 Vectors in C[sup(n)]......Page 20
2.2 Matrices......Page 34
2.3 Matrix Addition and Scalar Multiplication......Page 35
2.4 Summation Symbol......Page 36
2.5 Matrix Multiplication......Page 37
2.7 Square Matrices......Page 39
2.9 Invertible (Nonsingular) Matrices......Page 41
2.10 Special Types of Square Matrices......Page 42
2.11 Complex Matrices......Page 45
2.12 Block Matrices......Page 46
3.2 Basic Definitions, Solutions......Page 64
3.3 Equivalent Systems, Elementary Operations......Page 67
3.4 Small Square Systems of Linear Equations......Page 68
3.5 Systems in Triangular and Echelon Forms......Page 71
3.6 Gaussian Elimination......Page 74
3.7 Echelon Matrices, Row Canonical Form, Row Equivalence......Page 77
3.8 Gaussian Elimination, Matrix Formulation......Page 80
3.9 Matrix Equation of a System of Linear Equations......Page 84
3.10 Systems of Linear Equations and Linear Combinations of Vectors......Page 86
3.11 Homogeneous Systems of Linear Equations......Page 88
3.12 Elementary Matrices......Page 91
3.13 LU Decomposition......Page 94
4.2 Vector Spaces......Page 119
4.3 Examples of Vector Spaces......Page 120
4.4 Linear Combinations, Spanning Sets......Page 122
4.5 Subspaces......Page 124
4.6 Linear Spans, Row Space of a Matrix......Page 126
4.7 Linear Dependence and Independence......Page 128
4.8 Basis and Dimension......Page 131
4.9 Application to Matrices, Rank of a Matrix......Page 133
4.10 Sums and Direct Sums......Page 136
4.11 Coordinates......Page 137
5.2 Mappings, Functions......Page 171
5.3 Linear Mappings (Linear Transformations)......Page 174
5.4 Kernel and Image of a Linear Mapping......Page 176
5.5 Singular and Nonsingular Linear Mappings, Isomorphisms......Page 179
5.6 Operations with Linear Mappings......Page 180
5.7 Algebra A(V) of Linear Operators......Page 181
6.2 Matrix Representation of a Linear Operator......Page 202
6.3 Change of Basis......Page 206
6.4 Similarity......Page 210
6.5 Matrices and General Linear Mappings......Page 211
7.2 Inner Product Spaces......Page 233
7.3 Examples of Inner Product Spaces......Page 234
7.4 Cauchy–Schwarz Inequality, Applications......Page 236
7.5 Orthogonality......Page 238
7.6 Orthogonal Sets and Bases......Page 240
7.7 Gram–Schmidt Orthogonalization Process......Page 242
7.8 Orthogonal and Positive Definite Matrices......Page 244
7.9 Complex Inner Product Spaces......Page 246
7.10 Normed Vector Spaces (Optional)......Page 248
8.2 Determinants of Orders 1 and 2......Page 271
8.3 Determinants of Order 3......Page 272
8.5 Determinants of Arbitrary Order......Page 274
8.6 Properties of Determinants......Page 275
8.7 Minors and Cofactors......Page 276
8.8 Evaluation of Determinants......Page 277
8.9 Classical Adjoint......Page 278
8.10 Applications to Linear Equations, Cramer\'s Rule......Page 279
8.11 Submatrices, Minors, Principal Minors......Page 280
8.13 Determinants and Volume......Page 281
8.14 Determinant of a Linear Operator......Page 282
8.15 Multilinearity and Determinants......Page 283
9.1 Introduction......Page 299
9.2 Polynomials of Matrices......Page 300
9.3 Characteristic Polynomial, Cayley–Hamilton Theorem......Page 301
9.4 Diagonalization, Eigenvalues and Eigenvectors......Page 303
9.5 Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices......Page 306
9.6 Diagonalizing Real Symmetric Matrices and Quadratic Forms......Page 308
9.7 Minimal Polynomial......Page 310
9.8 Characteristic and Minimal Polynomials of Block Matrices......Page 312
10.2 Triangular Form......Page 332
10.3 Invariance......Page 333
10.5 Primary Decomposition......Page 334
10.6 Nilpotent Operators......Page 335
10.7 Jordan Canonical Form......Page 336
10.9 Rational Canonical Form......Page 337
10.10 Quotient Spaces......Page 338
11.2 Linear Functionals and the Dual Space......Page 356
11.4 Second Dual Space......Page 357
11.6 Transpose of a Linear Mapping......Page 358
12.2 Bilinear Forms......Page 366
12.4 Alternating Bilinear Forms......Page 367
12.5 Symmetric Bilinear Forms, Quadratic Forms......Page 368
12.6 Real Symmetric Bilinear Forms, Law of Inertia......Page 370
12.7 Hermitian Forms......Page 371
13.2 Adjoint Operators......Page 384
13.3 Analogy Between A(V) and C, Special Linear Operators......Page 385
13.5 Orthogonal and Unitary Operators......Page 387
13.7 Change of Orthonormal Basis......Page 388
13.9 Diagonalization and Canonical Forms in Inner Product Spaces......Page 389
13.10 Spectral Theorem......Page 390
Appendix A: Multilinear Products......Page 403
Appendix B: Algebraic Structures......Page 410
Appendix C: Polynomials over a Field......Page 418
Appendix D: Odds and Ends......Page 422
List of Symbols......Page 427
C......Page 428
I......Page 429
M......Page 430
R......Page 431
Z......Page 432