Corank
{{short description|Complementary of a rank}}
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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.{{cite book |last1=Loebl |first1=Martin |last2=Nešetřil |first2=Jaroslav |last3=Thomas |first3=Robin |title=A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek |date=11 October 2017 |publisher=Springer |isbn=978-3-319-44479-6 |url=https://books.google.com/books?id=aHg5DwAAQBAJ&dq=%22Corank%22+-wikipedia+math+nullspace&pg=PA572 |language=en}}
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. For a square matrix , the corank and nullity of 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 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.