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Corank

From Wikipedia, the free encyclopedia

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

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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

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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

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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.

References

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  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.