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.
Left nullspace of a matrix[]
The corank of an matrix is where 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 elements and matroid rank , the corank or nullity of the matroid is . 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.
- Linear algebra
- Matroid theory