Linear algebra is the mathematics of flat approximations
Lines, planes, rotations, and first-order models of curved systems all live inside vector spaces. From ancient Chinese counting rods through to modern matrix algorithms, linear algebra became the default shared language across geometry, the sciences, and computation.
The subject concerns linear equations and the structures that solve them systematically. It underpins modern geometry's definitions of lines, planes, and rotations, and functional analysis can be read as linear algebra on function spaces. Sciences and engineering use linear models constantly; even nonlinear problems lean on first-order approximations, where a manifold's derivative at a point is a linear map. Computing thrives on matrix algorithms for modeling and simulation.
History runs deep. Chapter Eight of the Chinese Nine Chapters on the Mathematical Art already shows rod-based elimination on systems of two to five equations. Descartes's 1637 coordinates made lines and planes into linear equations in Europe. Leibniz considered determinants in 1693; Cramer gave explicit formulas in 1750; Gauss refined elimination for geodesy. Nineteenth-century giants used the main techniques for a century without defining abstract vector spaces. Grassmann's 1844 Theory of Extension opened new ground; Sylvester coined "matrix" in 1848; Hamilton's 1843 quaternions popularized vectors; Cayley in 1856 introduced matrix multiplication and inverses, treating a matrix as a single letter. Peano gave a precise vector-space definition in 1888; by 1900 finite-dimensional linear transformations were a theory; the twentieth century abstracted further as computers demanded efficient elimination and decompositions.
Modern teaching prefers vector spaces over a field: an abelian group under addition with distributive, associative scalar multiplication. Vectors may be tuples, functions, polynomials, or matrices—linear algebra studies what they share. Linear maps preserve addition and scalar multiplication; choosing bases turns them into matrices. Until the nineteenth century the entry point was equations and arrays; the synthetic vector-space view is more general, not limited to finite dimensions, and conceptually cleaner once abstraction is accepted.
Source: Linear algebra
