homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
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Linear algebra over a skewfield $K$ is the study of the category $K$-Vect, that is the study of vector spaces over $K$. Sometimes one uses the term ‘$K$-linear algebra’ to mean an associative algebra (or similar) over $K$ (compare ‘$K$-linear map’).
Classical linear algebra is done over a real-closed field or an algebraically closed field of characteristic $0$; the latter is the simplest context, which is now pretty thoroughly understood. Traditionally, we take these fields to be the field of real numbers and the field of complex numbers, although arguably we only really use algebraic numbers. (From a constructive point of view, some of the classical material is valid only over discrete fields, so we must restrict attention to algebraic numbers, or to some discrete extension, for these results to hold in their classical form.)
Fancier linear algebra is done over incomplete fields and fields with positive characteristic (and constructively over nondiscrete fields). Sometimes a generalization to categories of finitely generated projectives over a ring is considered. In infinite dimensions one rarely studies purely algebraic version, which is considered as a linear algebra, but more often one equips them with topological structure, what enters the subject of functional analysis.
If one is interested in tensor products as well, then one gets a generalization called multilinear algebra: tensor algebra, tensors, exterior and symmetric algebras are some of the main characters in that theory. Study of determinants is important in the usual linear algebra but it is also closely related to the study of exterior algebras.
wikipedia linear algebra
Jean Dieudonné, Linear algebra and geometry, Translated from the French Houghton Mifflin Co., Boston, Mass. 1969, 207 pp
I. M. Gelʹfand, Lekcii po lineĭnoĭ algebre [2d ed.] Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951; Eng. transl: Lectures on linear algebra, Interscience Tracts in Pure and Applied Mathematics 9, 1961 ix+185 pp
A. I. Kostrikin, Ju. I. Manin, Lineĭnaya algebra i geometriya, Moskov. Gos. Univ., Moscow, 1980. 320 pp. MR82i:00008; Linear algebra and geometry, Translated from the second Russian edition by M. E. Alferieff. Algebra, Logic and Applications, 1. Gordon and Breach Science Publishers, New York, 1989. x+309 pp. MR91h:00008
Joel W. Robbin, Matrix algebra using MINImal MATlab
N.J. Higham, Functions of matrices. Theory and computation, Philadelphia, PA: SIAM. 425p., 2008
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