Corank

Short description: Complementary of a rank

In mathematics, corank is complementary to the concept of the rank of a mathematical object, and may refer to the dimension of the left nullspace of a matrix, the dimension of the cokernel of a linear transformation of a vector space, or the number of elements of a matroid minus its rank.

Left nullspace of a matrix

The corank of an [math]\displaystyle{ m\times n }[/math] matrix is [math]\displaystyle{ m-r }[/math] where [math]\displaystyle{ r }[/math] is the rank of the matrix. It is the dimension of the left nullspace and of the cokernel of the matrix.

Cokernel of a linear transformation

Generalizing matrices to linear transformations of vector spaces, the corank of a linear transformation is the dimension of the cokernel of the transformation, which is the quotient of the codomain by the image of the transformation.

Matroid

For a matroid with [math]\displaystyle{ n }[/math] elements and matroid rank [math]\displaystyle{ r }[/math], the corank or nullity of the matroid is [math]\displaystyle{ n-r }[/math]. In the case of linear matroids this coincides with the matrix corank. In the case of graphic matroids the corank is also known as the circuit rank or cyclomatic number.