Rank (linear algebra)

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The rank of a matrix A is the maximal number of linearly independent rows or columns of A. Since the column rank and the row rank are always equal, they are simply called the rank of A.[1] It is commonly denoted by either rk(A) or rank A.

The rank is 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 maximum rank of an m × n matrix is the lesser of m & n. A matrix that has a rank as large as possible is said to have full rank; otherwise, the matrix is rank deficient.


Alternative definitions

If one considers the matrix A as a linear map

with the rule

then the rank of A can also be defined as the dimension of the image of f (see linear map for a discussion of image and kernel). This definition has the advantage that it can be applied to any linear map without need for a specific matrix. The rank can also be defined as n minus the dimension of the kernel of f; the rank-nullity theorem states that this is the same as the dimension of the image of f.

The maximal number of linearly independent columns c^1,\dots,c^k of the m×n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of Fm generated by the columns of A, which is in fact just the image of A as a linear map).

Since the column rank and the row rank are the same, we can also define the rank of A as the dimension of the row space of A, or the number of rows r_1,\dots,r_k in a basis of the row space.

The rank can also be characterized as the decomposition rank: the minimum k such that A can be factored as A = CR, where C is an m×k matrix and R is a k×n matrix. Like the "dimension of image" characterization this can be generalized to a definition of the rank of a linear map: the rank of a linear map f from VW is the minimal dimension k of an intermediate space X such that f can be written as the composition of a map VX and a map XW. While this definition does not suggest an efficient manner to compute the rank (for which it is better to use one of the alternative definitions), it does allow to easily understand many of the properties of the rank, for instance that the rank of the transpose of A is the same as that of A.

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