Until the 19th century, linear algebra was introduced through systems of linear equations and matrices.
In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract.
Linear algebra is concerned with properties common to all vector spaces.
When a bijective linear map exists between two vector spaces (that is, every vector from the second space is associated with exactly one in the first), the two spaces are isomorphic.
Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group.
The mechanism of group representation became available for describing complex and hypercomplex numbers.
In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations.
The first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693.
The telegraph required an explanatory system, and the 1873 publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression.
Linear algebra is flat differential geometry and serves in tangent spaces to manifolds.