Corank

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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.^[1]

The corank of an ${\displaystyle m\times n}$ matrix is ${\displaystyle m-r}$ where ${\displaystyle r}$ 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 ${\displaystyle M}$, the corank and nullity of ${\displaystyle M}$ are equivalent.

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