In mathematics, corank may refer to any of several related concepts, complementary to the concept of the rank of a mathematical object:
The corank of an
m
×
n
matrix is
m
−
r
where
r
is the rank of the matrix. Equivalently, by the fundamental theorem of linear algebra, it is the dimension of the left nullspace of the matrix.
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.
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.
