Schedule
Term Spring 2027: 2027-01-19 through 2027-05-07.
| Unit | Topic focus |
|---|---|
| Week 1 — Vector spaces over a field, subspaces, and direct sums | the vector space axioms over an arbitrary field, fields other than the real numbers, subspaces and the subspace criterion, span and the subspace generated by a set, sums of subspaces, internal direct sums and complements, external direct sums |
| Week 2 — Linear independence, bases, and dimension | linear independence and dependence, the exchange lemma, bases as maximal independent sets and as minimal spanning sets, extending an independent set to a basis, dimension and its well-definedness, the dimension formula for a sum of subspaces, bases of infinite-dimensional spaces |
| Week 3 — Linear transformations, kernel and image, and rank-nullity | linear transformations and determination on a basis, kernel and image as subspaces, rank and nullity, the rank-nullity theorem, quotient spaces and the canonical projection, the first isomorphism theorem for vector spaces |
| Week 4 — Isomorphism, matrix representation, and change of basis | isomorphism and inverse transformations, dimension as a complete invariant, the vector space of linear maps, the matrix of a transformation relative to ordered bases, composition as matrix multiplication, change of basis, matrix equivalence and similarity |
| Week 5 — Inner products, norms, and the geometry they impose | real and complex inner products and conjugate symmetry, the induced norm, the Cauchy-Schwarz inequality, the triangle inequality, angle and orthogonality, the parallelogram law and which norms come from an inner product |
| Week 6 — Orthogonal sets, orthonormal bases, and Gram-Schmidt | orthogonal and orthonormal sets, orthogonality implies independence, coordinates against an orthonormal basis, the Gram-Schmidt process, the QR factorisation, Bessel’s inequality and Parseval’s identity |
| Week 7 — Orthogonal projections, best approximation, and least squares | the orthogonal projection onto a subspace, the best-approximation characterisation, projection matrices and their idempotence, the normal equations, least-squares fitting, projection onto a line and onto a column space |
| Week 8 — Orthogonal complements, adjoints, and the four fundamental subspaces | the orthogonal complement and its dimension, the direct sum decomposition of a space by a subspace and its complement, the adjoint of a linear map, self-adjoint and unitary maps, kernel and image of the adjoint, the four fundamental subspaces and their orthogonality relations |
| Week 9 — Bilinear forms, quadratic forms, congruence, and inertia | bilinear and sesquilinear forms, the matrix of a form and congruence, symmetric forms and their associated quadratic forms, diagonalising a quadratic form by completing the square, Sylvester’s law of inertia, definiteness and its tests |
| Week 10 — Eigenvalues, eigenvectors, and invariant subspaces | eigenvalues and eigenvectors of an operator, the characteristic polynomial, eigenspaces, invariant subspaces and restriction, the effect of the field on existence of eigenvalues, block triangular form from an invariant subspace |
| Week 11 — Similar matrices, diagonalization, and multiplicity | similarity as change of basis for an operator, similarity invariants including trace determinant and characteristic polynomial, algebraic and geometric multiplicity, the diagonalizability criterion, independence of eigenvectors for distinct eigenvalues, simultaneous diagonalization |
| Week 12 — Symmetric and Hermitian matrices and the spectral theorem | self-adjoint operators, real eigenvalues of Hermitian matrices, orthogonality of eigenvectors for distinct eigenvalues, the spectral theorem and orthogonal or unitary diagonalization, normal operators, the spectral decomposition as a sum of projections |
| Week 13 — Schur triangularization, the minimal polynomial, and Cayley-Hamilton | Schur’s triangularization theorem over the complex field, the minimal polynomial and its divisors, the Cayley-Hamilton theorem, minimal polynomial as a diagonalizability test, annihilating polynomials, computing with polynomial identities in an operator |
| Week 14 — Generalized eigenvectors, nilpotent operators, and primary decomposition | generalized eigenvectors and generalized eigenspaces, nilpotent operators and their index, cyclic chains and Jordan strings, the primary decomposition theorem, the dimension of a generalized eigenspace, the structure of a nilpotent operator |
| Week 15 — Jordan canonical form and the classification of operators | construction of the Jordan canonical form, uniqueness up to order of blocks, reading block sizes from ranks of powers, similarity decided by Jordan type, what the classification does and does not settle, the arc from vector spaces to canonical form |
The LMS remains authoritative for section logistics and graded details.