Linear algebra

Preface......Page 6
Contents......Page 10
PART I MATRICES AND VECTOR SPACES......Page 14
1.1 Matrices......Page 16
1.2 Addition and multiplication of matrices......Page 17
1.4 The transpose of a matrix......Page 18
1.5 Row and column operations......Page 19
1.6 Determinant and trace......Page 24
1.7 Minors and cofactors......Page 28
2.1 Examples and axioms......Page 32
2.2 Subspaces......Page 37
2.3 Linear independence......Page 40
2.4 Bases......Page 42
2.5 Coordinates......Page 47
2.6 Vector spaces over other fields......Page 51
PART II BILINEAR AND SESQUILINEAR FORMS......Page 58
3.1 The standard inner product......Page 60
3.2 Inner products......Page 62
3.3 Inner products over......Page 68
4.1 Bilinear forms......Page 74
4.2 Representation by matrices......Page 76
4.3 The base-change formula......Page 79
4.4 Sesquilinear forms over......Page 82
5.1 Orthonormal bases......Page 86
5.2 The Gram-Schmidt process......Page 88
5.3 Properties of orthonormal bases......Page 94
5.4 Orthogonal complements......Page 98
6.1 The Gram-Schmidt process revisited......Page 107
6.2 The leading minor test......Page 111
7.1 Quadratic forms......Page 119
7.2 Sylvester\'s law of inertia......Page 122
7.3 Examples......Page 126
7.4 Applications to surfaces......Page 130
7.5 Sesquilinear and Hermitian forms......Page 131
PART III LINEAR TRANSFORMATIONS......Page 138
8.1 Basics......Page 140
8.2 Arithmetic operations on linear transformations......Page 146
8.3 Representation by matrices......Page 148
9.1 Polynomials......Page 155
9.2 Evaluating polynomials......Page 156
9.3 Roots of polynomials over......Page 158
9.4 Roots of polynomials over other fields......Page 160
10.1 An example......Page 164
10.2 Eigenvalues and eigenvectors......Page 166
10.3 Upper triangular matrices......Page 168
11.1 The minimum polynomial......Page 175
11.2 The characteristic polynomial......Page 177
11.3 The Cayley-Hamilton theorem......Page 180
12.1 Diagonal matrices......Page 187
12.2 A criterion for diagonalizability......Page 189
12.3 Examples......Page 193
13.1 Orthogonal and unitary transformations......Page 200
13.2 From forms to transformations......Page 203
13.3 Eigenvalues and diagonalization......Page 207
13.4 Applications......Page 209
14.1 Jordan normal form......Page 216
14.2 Obtaining the Jordan normal form......Page 222
14.3 Applications......Page 225
14.4 Proof of the primary decomposition theorem......Page 229
Appendix A: A theorem of analysis......Page 235
Appendix B: Applications to quantum mechanics......Page 237
Index......Page 240