Advanced Linear Algebra

✍ Scribed by Hugo J. Woerdeman

• Publisher: CRC
• Year: 2016
• Tongue: English
• Leaves: 337
• Category: Library

✦ Table of Contents

Contents......Page 3
Preface......Page 7
Notation......Page 10
Figures......Page 13
Fields & Matrix Algebra......Page 14
The eld......Page 15
The eld axioms......Page 16
Field examples......Page 18
Matrix algebra over dierent elds......Page 24
Exercises......Page 33
Denition of a vector space......Page 40
Vector spaces of functions......Page 42
Subspaces and more examples of vector spaces......Page 45
Linear independence, span, and basis......Page 50
Coordinate systems......Page 58
Exercises......Page 61
Denition of a linear transformation......Page 68
Range and kernel of linear transformations......Page 70
Matrix representations of linear maps......Page 74
Exercises......Page 78
The Cayley{Hamilton theorem......Page 82
Jordan canonical form for nilpotent matrices......Page 84
An intermezzo about polynomials......Page 88
The Jordan canonical form......Page 91
The minimal polynomial......Page 95
Commuting matrices......Page 97
Systems of linear dierential equations......Page 100
Functions of matrices......Page 103
The resolvent......Page 111
Exercises......Page 113
Inner products and norms......Page 122
Orthogonal and orthonormal sets and bases......Page 132
The adjoint of a linear map......Page 135
Unitary matrices, QR, and Schur triangularization......Page 138
Normal and Hermitian matrices......Page 141
Singular value decomposition......Page 145
Exercises......Page 150
The Cartesian product......Page 160
The quotient space......Page 162
The dual space......Page 170
Multilinear maps and functionals......Page 179
The tensor product......Page 181
Anti-symmetric and symmetric tensors......Page 192
Exercises......Page 202
How to use Linear Algebra......Page 208
Matrices you can't write down, but would still like to use......Page 209
Algorithms based on matrix vector products......Page 211
Why use matrices when computing roots of polynomials?......Page 216
How to nd functions with linear algebra?......Page 222
How to deal with incomplete matrices......Page 230
Solving millennium prize problems with linear algebra......Page 235
How secure is RSA encryption?......Page 242
Quantum computation and positive maps......Page 245
Exercises......Page 251
Bibliography for Chapter 7......Page 258
Answers......Page 262
Chapter 2......Page 268
Chapter 3......Page 279
Chapter 4......Page 285
Chapter 5......Page 300
Chapter 6......Page 315
Chapter 7......Page 326
How to start own Research Project......Page 260
Biblio......Page 336