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1 Linear algebra: key concepts
1.1 Vector spaces
1.2 Subspaces
1.3 Bases
1.4 Linear maps
2 Sums and quotients
2.1 Sums of subspaces
2.2 Direct sums
2.3 Quotients
3 Polynomials, operators and matrices
3.1 Polynomials
3.2 Linear operators, matrices and polynomials
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
4.4 Jordan normal form
5 Symmetric bilinear forms and quadratic forms
5.1 Bilinear forms and matrices
5.2 Symmetric bilinear forms
5.3 Application: Quadratic forms
Convention. In this chapter, all vector spaces are over the same field F unless we say otherwise.
Definition. Let V1,…,Vk≤V. The sum V1+⋯+Vk is the set
V1+⋯+Vk:={v1+⋯+vk|vi∈Vi,1≤i≤k}.
Proposition 2.1. Let V1,…,Vk≤V. Then
(1) V1+⋯+Vk≤V.
(2) If W≤V and V1,…,Vk≤W then V1,…,Vk≤V1+⋯+Vk≤W.