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2. Rank–nullity theorem - Wikipedia

   en.wikipedia.org/wiki/Rank–nullity_theorem

   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 ...

3. Rank (linear algebra) - Wikipedia

   en.wikipedia.org/wiki/Rank_(linear_algebra)

   A matrix that has rank min(m, n) is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. f is injective (or "one-to-one") if and only if A has rank n (in this case, we say that A has full column rank). f is surjective (or "onto") if and only if A has rank m (in this case, we say that A has full row ...

4. Rank (graph theory) - Wikipedia

   en.wikipedia.org/wiki/Rank_(graph_theory)

   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 ...

5. Classification theorem - Wikipedia

   en.wikipedia.org/wiki/Classification_theorem

   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 ...

6. Row and column spaces - Wikipedia

   en.wikipedia.org/wiki/Row_and_column_spaces

   The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. [9] The rank of a matrix is also equal to the dimension of the column space.

7. Nullity (graph theory) - Wikipedia

   en.wikipedia.org/wiki/Nullity_(graph_theory)

   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.

8. Quotient space (linear algebra) - Wikipedia

   en.wikipedia.org/wiki/Quotient_space_(linear...

   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).

9. Rank of a group - Wikipedia

   en.wikipedia.org/wiki/Rank_of_a_group

   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] The rank problem is undecidable for word hyperbolic groups. [15]