Corank

Complementary of a rank

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Corank" – news · newspapers · books · scholar · JSTOR (February 2024) (Learn how and when to remove this message)

|

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]

## Left nullspace of a matrix

[edit]

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

## Cokernel of a linear transformation

[edit]

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

[edit]

For a matroid with  n {\displaystyle n} elements and matroid rank r {\displaystyle r} , the corank or nullity of the matroid is  n − r {\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.

## References

[edit]

1. ^ Loebl, Martin; Nešetřil, Jaroslav; Thomas, Robin (11 October 2017). A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek. Springer. ISBN 978-3-319-44479-6.