Rank (linear algebra)

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This is the same as the dimension of the space spanned by its rows. It is a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.

The rank is commonly denoted rank(A) or rk(A); sometimes the parentheses are not written, as in rank A.

In this section we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these.

The column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A.

A fundamental result in linear algebra is that the column rank and the row rank are always equal. (Two proofs of this result are given in Proofs that column rank = row rank below.) This number (i.e., the number of linearly independent rows or columns) is simply called the rank of A.

A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank deficient if it does not have full rank.

The rank is also the dimension of the image of the linear transformation that is given by multiplication by A. More generally, if a linear operator on a vector space (possibly infinite-dimensional) has finite-dimensional image (e.g., a finite-rank operator), then the rank of the operator is defined as the dimension of the image.

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