By homotopy linear algebra we mean the study of linear functors between slices of the infinity-category of infinity-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into infinity-categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality a la Baez, Hoffnung and Walker compatible with this duality. We needed these results to support our work on incidence algebras and Mobius inversion over infinity-groupoids; we hope that they can also be of independent interest.