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The first four axioms mean that V is an abelian group under addition. Elements of a vector space may have various nature; for example, they can be sequences, functions, polynomials or matrices.
Linear algebra is concerned with properties common to all vector mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring. The matrix product is designed for representing the composition of linear maps that are represented by matrices.