Unit notes
The units below are the teaching spine of the course:
- Week 1 — Vector spaces over a field, subspaces, and direct sums
- Week 2 — Linear independence, bases, and dimension
- Week 3 — Linear transformations, kernel and image, and rank-nullity
- Week 4 — Isomorphism, matrix representation, and change of basis
- Week 5 — Inner products, norms, and the geometry they impose
- Week 6 — Orthogonal sets, orthonormal bases, and Gram-Schmidt
- Week 7 — Orthogonal projections, best approximation, and least squares
- Week 8 — Orthogonal complements, adjoints, and the four fundamental subspaces
- Week 9 — Bilinear forms, quadratic forms, congruence, and inertia
- Week 10 — Eigenvalues, eigenvectors, and invariant subspaces
- Week 11 — Similar matrices, diagonalization, and multiplicity
- Week 12 — Symmetric and Hermitian matrices and the spectral theorem
- Week 13 — Schur triangularization, the minimal polynomial, and Cayley-Hamilton
- Week 14 — Generalized eigenvectors, nilpotent operators, and primary decomposition
- Week 15 — Jordan canonical form and the classification of operators