Short answers, pulled from the story.
Linear algebra is the branch of mathematics concerning linear equations, linear maps, and their representations in vector spaces and through matrices. It is central to almost all areas of mathematics and is used across the sciences and engineering to model natural phenomena and compute with those models.
James Joseph Sylvester introduced the term matrix in 1848. The word is Latin for womb. Arthur Cayley later introduced matrix multiplication and the inverse matrix in 1856, treating a matrix as a single aggregate object denoted by one letter.
The procedure now called Gaussian elimination appears in the ancient Chinese mathematical text The Nine Chapters on the Mathematical Art, in Chapter Eight: Rectangular Arrays. Its use is illustrated in eighteen problems, with two to five equations each, originally solved using counting rods.
A vector space over a field is a set equipped with vector addition and scalar multiplication that satisfy axioms including associativity, commutativity, a zero vector, additive inverses, distributivity, and an identity scalar. Its elements, called vectors, may be tuples, sequences, functions, polynomials, or matrices.
An eigenvector of a linear endomorphism is a nonzero vector that the map sends to a scalar multiple of itself, and that scalar is the eigenvalue. The eigenvalues are the roots of the characteristic polynomial, a monic polynomial of degree n for a space of dimension n, giving at most n eigenvalues.
Linear algebra underpins functional analysis, quantum mechanics, Fourier analysis, and scientific computation, with BLAS and LAPACK among its best known implementations. It is used in weather forecasting, where the atmosphere is divided into cells of roughly 100 km, and in fluid mechanics, computational fluid dynamics, the Navier-Stokes equations, and power systems analysis.