Mn×p

Mn×p denotes the set of all n-by-p matrices with entries from a field F, typically the real numbers R or the complex numbers C. It forms a vector space over F under standard matrix addition and scalar multiplication. The zero matrix is the additive identity, and every element has an additive inverse.

The dimension of Mn×p as a vector space is n·p. A convenient basis is the family of matrices Eij that have a 1 in position (i, j) and 0 elsewhere, for i = 1,…,n and j = 1,…,p. Consequently Mn×p is naturally isomorphic, as a vector space, to F^n ⊗ F^p (and to F^(n·p)).

Matrix multiplication is defined only for compatible sizes. If A ∈ M(m×n) and B ∈ M(n×p), their product AB lies in M(m×p). The set Mn×p itself is not closed under multiplication unless p = n and one restricts to square matrices, in which case Mn denotes the algebra of all n×n matrices with standard addition and multiplication.

Mn×p also represents linear maps from F^p to F^n: each matrix A defines a linear transformation x ↦ Ax. Thus Mn×p is isomorphic to the space of linear maps Hom(F^p, F^n).

Common subspaces include the row space and column space of a particular matrix, as well as subspaces defined by linear relations among its entries. The rank of individual matrices is a fundamental, but separate, notion from the structure of the set Mn×p. Special cases: when n = p, Mn×p is the standard matrix algebra M_n; when p ≠ n, it describes linear transformations between spaces of different dimensions.