Tag
Linear Algebra
Linear algebra is the study of vector spaces and the linear maps between them. A matrix is a coordinate representation of such a map, and almost every computational subject in this site's orbit is linear algebra wearing a costume: the solution set of a linear system of ODEs, the conservation laws of a reaction network, the attention scores of a transformer. Writing here treats the maps as the objects and the matrices as their shadows.
Blog
July 13, 2026
Linear Algebra I: Vector Spaces and Linear Maps
A proof-based reconstruction of linear algebra from the axioms: vector spaces, bases and dimension by Steinitz exchange, linear maps and rank-nullity, matrices as coordinate representations, the four fundamental subspaces, and the affine solution set of a linear system. The application is stoichiometry: balancing a reaction is a search for integer vectors in a null space, conservation laws are a left null space, and an ICE table is an affine set cut by a nonlinear equilibrium variety. Part I of a six-part lecture series converging on the singular value decomposition and geometric algebra.
April 12, 2026
Analysis on Manifolds II: Exterior Algebra
The algebraic machinery behind differential forms: dual spaces, multilinear alternating maps, the wedge product, bases and dimension of Λᵏ(V*), determinants as top forms, the interior product, and the Hodge star. Part II of a five-part lecture series on differential forms and the generalised Stokes theorem.