Advanced Linear Algebra

Rebuild linear algebra as a theory rather than a matrix technique: work with vector spaces over an arbitrary field, prove the structural results that make dimension, rank, and isomorphism mean what they mean, add geometry through inner products and orthogonality, and end by classifying an operator up to similarity — seeing throughout that a matrix is the shadow a linear map casts once a basis has been chosen.

3 credits · in_person · Spring 2027

What you will learn

State the vector space axioms over an arbitrary field and verify or refute that a given set with given operations is a vector space.
Recognise subspaces, describe the subspace spanned by a set, and decide when a sum of subspaces is direct.
Prove and apply the exchange lemma, and use it to show that dimension is well defined.
Characterise a basis as a maximal independent set and as a minimal spanning set, and construct bases by extension and by reduction.
Apply the dimension formula for the sum of two subspaces and use it to decide questions about intersections and complements.
Determine a linear transformation from its values on a basis, and compute its kernel, image, rank, and nullity.
Prove the rank-nullity theorem, derive it from the first isomorphism theorem, and construct quotient spaces.
Decide when two vector spaces are isomorphic, write the matrix of a map relative to ordered bases, and perform a change of basis.
Distinguish matrix equivalence from similarity by what each relation is allowed to change and what each preserves.
Work with inner products over the real and complex fields, derive the Cauchy-Schwarz and triangle inequalities, and use the norms they induce.
Construct orthonormal bases by Gram-Schmidt, and use orthogonal sets to compute coordinates without solving a system.
Compute orthogonal projections onto a subspace, characterise them by the best-approximation property, and solve least-squares problems.
Use orthogonal complements and the adjoint of a linear map, and relate the four fundamental subspaces of a map to one another.
Classify bilinear and quadratic forms up to congruence, diagonalise a quadratic form, and apply Sylvester’s law of inertia.
Compute eigenvalues, eigenvectors, and characteristic polynomials, and identify invariant subspaces of an operator.
Decide whether an operator is diagonalizable by comparing algebraic and geometric multiplicities, and explain what similarity preserves.
State and prove the spectral theorem for symmetric and Hermitian matrices, and recognise normal operators.
Apply Schur triangularization, the minimal polynomial, and the Cayley-Hamilton theorem to an operator.
Decompose a space into generalized eigenspaces, analyse nilpotent operators, and construct the Jordan canonical form of an operator.
Use the Jordan form to decide whether two matrices are similar, and state exactly what it classifies.

How the course runs

Advanced Linear Algebra meets as three lecture hours a week on days still to be assigned through Spring 2027, 2027-01-19 to 2027-05-07, across 15 units. This site is a draft. Zero student cost, with the course site as the primary source of truth and every reading it points to openly available online. The reference text named on the department syllabus is treated as an optional reference and is never required to be purchased; nothing from it is reproduced here. The learning management system stays authoritative for section logistics, dates, and graded details.

Start here

Begin with Week 1 — Vector spaces over a field, subspaces, and direct sums, then follow the unit list in the sidebar.

How to use this site

Everything you need is on this site, at no cost: unit notes, the syllabus summary, the schedule, and supporting resources. The LMS remains authoritative for section logistics and graded details.