Here and in the sequel we identify Rⁿ with the set of "columns" or n-by-1 matrices. For every linear map f : Rⁿ → R^m there exists a unique m-by-n matrix A such that f(x) = Ax for all x in Rⁿ. We say that the matrix A "represents" the linear map f. Now if the k-by-m matrix B represents another linear map g : R^m → R^k, then the linear map g o f is represented by BA. This follows from the above-mentioned associativity of matrix multiplication.

More generally, a linear map from an n-dimensional vector space to an m-dimensional vector space is represented by an m-by-n matrix, provided that bases have been chosen for each.

A matrix where one of the dimensions equals one is often called a vector, and interpreted as an element of real coordinate space. An matrix (one column and m rows) is called a column vector and an matrix (one row and n columns) is called a row vector.

Definitions and notations

The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (written ) and m and n are called its dimensions. The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an m-by-n matrix has an order of (order meaning size).

Almost always capital letters are used to denote matrices with the corresponding lower case letter with two indices representing the entries. For example the entry of a matrix A that lies in the i-th row and the j-th column is written as a_i,j and called the i,j entry or (i,j)-th entry of A. Alternative notations for that entry are A[i,j] or A_i,j. The row is always noted first, then the column.

We often write or to define an matrix A. In this case the entries a_i,j are defined separately for all integers and . In some programming languages the numbering of rows and colums starts at zero. Texts, which make use of such a language extensively, frequently follow that convention, so we have and .