1 Linear algebra: key concepts
1.1 Vector spaces
1.2 Subspaces
1.3 Bases
1.3.1 Useful facts
1.4 Linear maps
2 Sums and quotients
2.1 Sums of subspaces
2.2 Direct sums
2.2.1 Induction from two summands
2.2.2 Direct sums, bases and dimension
2.2.3 Complements
2.3 Quotients
3 Polynomials, operators and matrices
3.1 Polynomials
3.2 Linear operators, matrices and polynomials
3.2.1 Linear operators and matrices
3.2.2 Polynomials in linear operators and matrices
3.3 The minimum polynomial
3.4 Eigenvalues and the characteristic polynomial
3.5 The Cayley–Hamilton theorem
4 The structure of linear operators
4.1 On normal forms
4.2 Invariant subspaces
4.3 Jordan decomposition
4.3.1 Powers of operators and Fitting’s Lemma
4.3.2 Generalised eigenspaces
4.4 Jordan normal form
4.4.1 Jordan blocks
4.4.2 Jordan normal form
5 Symmetric bilinear forms and quadratic forms
5.1 Bilinear forms and matrices
5.2 Symmetric bilinear forms
5.2.1 Rank and radical
5.2.2 Classification of symmetric bilinear forms
5.2.3 Sylvester’s Theorem
5.3 Application: Quadratic forms