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Basics of Algebra, Topology, and Differential Calculus

Contents
1 Introduction
9
2 Vector Spaces, Bases, Linear Maps 11
2.1 Groups, Rings, and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Linear Independence, Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Bases of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Matrices and Linear Maps 45
3.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Haar Basis Vectors and a Glimpse at Wavelets . . . . . . . . . . . . . . . . . 61
3.3 The Eect of a Change of Bases on Matrices . . . . . . . . . . . . . . . . . . 77
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
and Linear Forms . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Hyperplanes and Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4 Transpose of a Linear Map and of a Matrix . . . . . . . . . . . . . . . . . . . 108
4.5 The Four Fundamental Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 Determinants 123
5.1 Permutations, Signature of a Permutation . . . . . . . . . . . . . . . . . . . 123
5.2 Alternating Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Denition of a Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.4 Inverse Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . . 136
5.5 Systems of Linear Equations and Determinants . . . . . . . . . . . . . . . . 140
5.6 Determinant of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.7 The Cayley{Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 141
3
4 Direct Sums, The Dual Space, Duality 81
4.1 Sums, Direct Sums, Direct Products . . . . . . . . . . . . . . . . . . . . . . 81
4.2 The Dual Space E
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