Mm×n

Mm×n denotes the set of all m-by-n matrices with entries drawn from a field F, typically the real numbers R or the complex numbers C. An element A in Mm×n has entries aij for i = 1,…,m and j = 1,…,n, arranged in a rectangular array with m rows and n columns.

As a mathematical object, Mm×n forms a vector space over F under standard matrix addition and scalar multiplication. Its dimension is m n, and a common basis is the set of matrices Eij, which have a 1 in position (i,j) and 0 elsewhere. The matrix A represents a linear map from F^n to F^m when bases are chosen for these vector spaces; applying A to a column vector in F^n yields a vector in F^m.

Matrix operations include addition, scalar multiplication, and multiplication by matrices of compatible sizes. If A is m×n and B is n×p, their product AB is defined and results in an m×p matrix. Transpose provides a related operation, mapping Mm×n to Mn×m. In general, the set Mm×n is not closed under multiplication, so it does not form a ring unless m = n; the special case Mn(F) (the set of all n×n matrices) forms a ring and an algebra over F.

Mm×n is central in linear algebra for representing linear transformations between finite-dimensional vector spaces, solving systems of linear equations, and describing matrix factorizations, bases, and rank-related properties.