The corank of an m\times n matrix is m-r where r is the rank of the matrix. It is the dimension of the left nullspace and of the cokernel of the matrix.
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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.
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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.
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Umbilic catastrophes are examples of corank 2 catastrophes. They can be observed in optics in the focal surfaces created by light reflecting off a surface in three dimensions and are intimately connected with the geometry of nearly spherical surfaces: umbilical point. Thom proposed that the hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical umbilic modeled the creation of hair-like structures.
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The circuit rank of a graph may be described using matroid theory as the corank of the graphic matroid of .. Using the greedy property of matroids, this means that one can find a minimum set of edges that breaks all cycles using a greedy algorithm that at each step chooses an edge that belongs to at least one cycle of the remaining graph. Alternatively, a minimum set of edges that breaks all cycles can be found by constructing a spanning forest of and choosing the complementary set of edges that do not belong to the spanning forest.
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The dual notion of a uniform module is that of a hollow module: a module M is said to be hollow if, when N1 and N2 are submodules of M such that N_1+N_2=M, then either N1 = M or N2 = M. Equivalently, one could also say that every proper submodule of M is a superfluous submodule. These modules also admit an analogue of uniform dimension, called co-uniform dimension, corank, hollow dimension or dual Goldie dimension. Studies of hollow modules and co-uniform dimension were conducted in , , , and . The reader is cautioned that Fleury explored distinct ways of dualizing Goldie dimension.
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More generally, a matroid is called graphic whenever it is isomorphic to the graphic matroid of a graph, regardless of whether its elements are themselves edges in a graph. The bases of a graphic matroid M(G) are the spanning forests of G, and the circuits of M(G) are the simple cycles of G. The rank in M(G) of a set X of edges of a graph G is r(X)=n-c where n is the number of vertices in the subgraph formed by the edges in X and c is the number of connected components of the same subgraph. The corank of the graphic matroid is known as the circuit rank or cyclomatic number.
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In graph theory, the circuit rank (or cyclomatic number) of a graph is the corank of the associated graphic matroid; it measures the minimum number of edges that must be removed from the graph to make the remaining edges form a forest.. Several authors have studied the parameterized complexity of graph algorithms parameterized by this number... In linear algebra, the rank of a linear matroid defined by linear independence from the columns of a matrix is the rank of the matrix,. and it is also the dimension of the vector space spanned by the columns. In abstract algebra, the rank of a matroid defined from sets of elements in a field extension L/K by algebraic independence is known as the transcendence degree..
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It is also possible to construct a minimum-size set of edges that breaks all cycles efficiently, either using a greedy algorithm or by complementing a spanning forest. The circuit rank can be explained in terms of algebraic graph theory as the dimension of the cycle space of a graph, in terms of matroid theory as the corank of a graphic matroid, and in terms of topology as one of the Betti numbers of a topological space derived from the graph. It counts the ears in an ear decomposition of the graph, forms the basis of parameterized complexity on almost-trees, and has been applied in software metrics as part of the definition of cyclomatic complexity of a piece of code. Under the name of cyclomatic number, the concept was introduced by Gustav Kirchhoff.
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