Summary
This is a first-year university course in linear algebra. It asks how far one linear map on a finite-dimensional vector space can be simplified by choosing a good basis. It then studies spaces with extra structure: dual spaces, inner products and bilinear forms. It suits students who already know vector spaces, matrices, determinants and eigenvalues.
Chapters 1-2: The Minimum Polynomial and Direct Sums of Subspaces
These chapters set up the main tools. A sum of subspaces is direct when every vector splits in exactly one way. This is the same as the pieces meeting only in , and it means dimensions add. Direct sums are linked to projections, which are exactly the idempotent maps (). The next idea is an invariant subspace, one that a map sends into itself. Splitting a space into invariant subspaces makes the matrix of block diagonal. The Primary Decomposition Theorem uses the factorised minimum and characteristic polynomials to split the space into kernels of powers of irreducible factors. It also gives a test for diagonalizability: the minimum polynomial must be a product of distinct linear factors. Commuting diagonalizable maps can be diagonalized together. Nilpotent maps are introduced here too.
Chapters 3-4: Reduction to Triangular Form and Reduction to Jordan Form
These chapters build a basis in which a nilpotent map is strictly upper triangular. They use this to triangularize any map whose eigenvalues lie in the field. The Jordan decomposition writes , where is diagonalizable, is nilpotent and the two commute. The kernel chain of a nilpotent map then gives Jordan blocks, and the exponents in the characteristic and minimum polynomials fix the number and sizes of the blocks. Worked examples find a Jordan basis and use it to solve linear differential systems .
Chapter 5: The Rational and Classical Forms
When eigenvalues lie outside the field, Jordan form fails. Quotient spaces and cyclic subspaces replace it. The annihilator of a vector gives a companion matrix. The Cyclic Decomposition Theorem and the elementary divisors then give the rational canonical form, and the classical form comes from it. The classical form reduces to Jordan form when all eigenvalues are in the field.
Chapter 6: Dual Spaces
This chapter covers linear functionals, the dual basis, the transpose of a map and annihilators. It also covers the identification of a finite-dimensional space with its bidual, and the equality of row rank and column rank.
Chapters 7-8: Inner Product Spaces and Orthogonal Direct Sums
These chapters cover inner products, norms, the Cauchy-Schwarz inequality, Gram-Schmidt and orthonormal expansions. They also cover adjoints and orthogonal complements, with . Self-adjoint projections lead to the spectral resolution. A complex map is ortho-diagonalizable exactly when it is normal. A real map is ortho-diagonalizable exactly when it is self-adjoint (symmetric matrices).
Chapter 9: Bilinear and Quadratic Forms
The last chapter covers the matrix of a bilinear form after a change of basis and congruence of matrices. It then covers quadratic forms, Sylvester's law of inertia and positive definiteness. It also describes skew-adjoint and real normal maps by block forms.
Key ideas:
- A direct sum gives unique decompositions, and dimensions add across it.
- Projections are exactly the idempotent maps, with .
- Primary decomposition splits a space into invariant kernels taken from the minimum polynomial.
- A map is diagonalizable if and only if its minimum polynomial has distinct linear factors.
- Jordan form exists when all eigenvalues lie in the field. Its block sizes come from the kernel chains of .
- Cyclic subspaces and companion matrices give rational canonical form over any field.
- Adjoints, orthogonal complements and the spectral theorem tie symmetric and normal maps to orthonormal eigenbases.
- Real symmetric matrices are congruent to , and and are fixed (Sylvester).