Rank (linear Algebra)

Rank (linear Algebra)

The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A. Equivalently, 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 result of fundamental importance in linear algebra is that the column rank and the row rank are always equal (see below for proofs). This number (i.e. the number of linearly independent rows or columns) is simply called the rank of A. It is commonly denoted by either rk(A) or rank A. Since the column vectors of A are the row vectors of the transpose of A (denoted here by AT), column rank of A equals row rank of A is equivalent to saying that the rank of a matrix is equal to the rank of its transpose, i.e. rk(A) = rk(AT).

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

The rank of an m × n matrix cannot be greater than m nor n. A matrix that has a rank as large as possible is said to have full rank; otherwise, the matrix is rank deficient.