高清版本 Notes on Linear Algebra  by Peter J. Cameron
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书籍目录




Contents
1 Vector spaces 3
1.1 Definitions . . . . . . . . . . . . . . . . . . . . .. . . . . . 3
1.2 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Row and column vectors . . . . . . . . .  . . . . . . . 9
1.4 Change of basis . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Subspaces and direct sums . . . . . . . . . . . . . . 13
2 Matrices and determinants 15
2.1 Matrix algebra . . . . . . . . . . . . . . . . . . .  . . . . 15
2.2 Row and column operations . . . . . . . . . . . . . . 16
2.3 Rank . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . 20
2.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Calculating determinants . . . . . . . . . . . .  . . . . 25
2.6 The Cayley–Hamilton Theorem . . . . . . . . . .  . . 29
3 Linear maps between vector spaces 33
3.1 Definition and basic properties . . . . . . . . . . . . . 33
3.2 Representation by matrices . . . . . . . . . . . . . . . 35
3.3 Change of basis . . . . . . . . . . . . . . . .  . . . . . . . 37
3.4 Canonical form revisited . . . . . . . . . . . . . . . . . . 39
4 Linear maps on a vector space  . . . .. . . . . . .. . . . . 41
4.1 Projections and direct sums . . . . . . .. . . . . . . . . 41
4.2 Linear maps and matrices . . . . . . . . . . . . . . . . . 43
4.3 Eigenvalues and eigenvectors . . . . . . . . . . . .. . . 44
4.4 Diagonalisability . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Characteristic and minimal polynomials . . . . . . . . 48
4.6 Jordan form . . . . . . . . . . . . . . . . . . . . . . .  . . . . 51
4.7 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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5 Linear and quadratic forms 55
5.1 Linear forms and dual space . . . . . . . . . . . . . . . . . . . . 55
5.1.1 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.2 Change of basis . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.1 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . 58
5.2.2 Reduction of quadratic forms . . . . . . . . . . . . . . . . 60
5.2.3 Quadratic and bilinear forms . . . . . . . . . . . . . . . . 62
5.2.4 Canonical forms for complex and real forms . . . . . . . 64
6 Inner product spaces 67
6.1 Inner products and orthonormal bases . . . . . . . . . . . . . . . 67
6.2 Adjoints and orthogonal linear maps . . . . . . . . . . . . . . . . 70
7 Symmetric and Hermitian matrices 73
7.1 Orthogonal projections and orthogonal decompositions . . . . . . 73
7.2 The Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . 75
7.3 Quadratic forms revisited . . . . . . . . . . . . . . . . . . . . . . 77
7.4 Simultaneous diagonalisation . . . . . . . . . . . . . . . . . . . . 78
8 The complex case 81
8.1 Complex inner products . . . . . . . . . . . . . . . . . . . . . . . 81
8.2 The complex Spectral Theorem . . . . . . . . . . . . . . . . . . . 82
8.3 Normal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9 Skew-symmetric matrices 85
9.1 Alternating bilinear forms . . . . . . . . . . . . . . . . . . . . . . 85
9.2 Skew-symmetric and alternating matrices . . . . . . . . . . . . . 86
9.3 Complex skew-Hermitian matrices . . . . . . . . . . . . . . . . . 88
A Fields and vector spaces 89
B Vandermonde and circulant matrices 93
C The Friendship Theorem 97
D Who is top of the league? 101
E Other canonical forms 105
F Worked examples 107