Linear Algebra

Fields......Page 2
Vector spaces......Page 4
Matrices......Page 7
Linear systems and matrices......Page 10
Resolution of linear systems......Page 12
Calculating the inverse of a matrix using elementary row and column operations......Page 15
Linear combination and linear independence in general vector spaces......Page 16
Bases......Page 18
Dimension......Page 20
Operations on subspaces......Page 23
Image and kernel of a matrix......Page 26
The change of basis formula......Page 30
Linear transformations......Page 31
Linear transformations and matrices......Page 35
Kernel and image of a linear transformation......Page 38
Rank-nullity theorem for linear transformations......Page 39
Dual space......Page 41
Determinants 1 : multilinear forms......Page 46
Determinants 2 : the symmetric group Sn......Page 48
Determinants 3 : alternating n-linear forms on a n-dimensional vector space......Page 51
Determinants 4 : determinant of an endomorphism......Page 54
Determinants 5 : determinant of a square matrix......Page 55
Determinants 6 : further properties of determinants......Page 58
Eigenvalues, eigenvectors and eigenspaces......Page 64
The characteristic polynomial......Page 65
Diagonalization......Page 68
Triangularization......Page 71
Some properties of polynomials......Page 74
Polynomials of matrices......Page 77
Cayley-Hamilton theorem and minimal polynomial......Page 80
Characteristic subspaces......Page 84
Jordan-Chevalley decomposition......Page 85
Jordan normal/canonical form......Page 87
Matrix of a bilinear form......Page 90
Symmetric bilinear form vs quadratic forms......Page 92
Non-degenerate and definite forms......Page 93
Orthogonals......Page 94
Orthogonal bases......Page 96
Real vector spaces......Page 98
Inner products......Page 99
Orthonormal bases......Page 101
A little bit of topology......Page 104
Isometries (a.k.a. orthogonal transformations)......Page 106
Adjoint of a linear transformation......Page 109
The spectral theorem : diagonalization of self-adjoint transformations......Page 110
The case of C-vector spaces......Page 112