1 Notation and terminology 1
2 Fields 5
3 Vector spaces over a field 17
4 Algebras over a field 33
5 Linear independence and dimension 49
6 Linear transformations 79
7 The endomorphism algebra of a vector space 99
8 Representation of linear transformations by matrices 117
9 The algebra of square matrices 131
10 Systems of linear equations 169
11 Determinants 199
12 Eigenvalues and eigenvectors 229
13 Krylov subspaces 267
14 The dual space 285
15 Inner product spaces 299
16 Orthogonality 325
17 Selfadjoint Endomorphisms 349
18 Unitary and Normal endomorphisms 369
19 Moore-Penrose pseudoinverses 389
20 Bilinear transformations and forms 399
A Summary of Notation 423
Index 427