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Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and; the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...
A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank-deficient if it does not have full rank. The rank deficiency of a matrix is the difference between the lesser of the number of rows and columns, and ...
The circuit rank of a hypergraph can be derived by its Levi graph, with the same circuit rank but reduced to a simple graph. = (+) + where g is the degree sum, e is the number of edges in the given graph, v is the number of vertices, and c is the number of connected components.
Equivalently, the rank of a graph is the rank of the oriented incidence matrix associated with the graph. [2] Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula m − n + c, where n and c are as above and m is the number of edges in the graph. The nullity is equal to the first Betti ...
The nullity of M is given by m − n + c, where, c is the number of components of the graph and n − c is the rank of the oriented incidence matrix. This name is rarely used; the number is more commonly known as the cycle rank, cyclomatic number, or circuit rank of the graph. It is equal to the rank of the cographic matroid of the graph.
An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T). The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).
The equivalence problem is "given two objects, determine if they are equivalent". A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it. (A combination of invariant values is realizable if there in fact exists an object whose invariants take on the ...
The rank problem is decidable for finite groups and for finitely generated abelian groups. The rank problem is decidable for finitely generated nilpotent groups. The reason is that for such a group G, the Frattini subgroup of G contains the commutator subgroup of G and hence the rank of G is equal to the rank of the abelianization of G. [14]