Corank

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 $$m\times n$$ matrix is $$m-r$$ where $$r$$ is the rank of the matrix. It is the dimension of the left nullspace and of the cokernel of the matrix. For a square matrix $$M$$, the corank and nullity of $$M$$ are equivalent.

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 $$n$$ elements and matroid rank $$r$$, the corank or nullity of the matroid is $$n-r$$. 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.