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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 ...
As a consequence, a rank-k matrix can be written as the sum of k rank-1 matrices, but not fewer. The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix. (This is the rank–nullity theorem.) If A is a matrix over the real numbers then the rank of A and the rank of its corresponding Gram matrix are equal.
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. The dimension of the null space is called the nullity of the matrix, and is related to the rank by the following equation:
The fact that the rank is finite is Francesco Severi's theorem of the base; the rank is the Picard number of V, often denoted ρ(V). The elements of finite order are called Severi divisors, and form a finite group which is a birational invariant and whose order is called the Severi number .
Principal axis theorem (linear algebra) Rank–nullity theorem (linear algebra) Rouché–Capelli theorem (Linear algebra) Sinkhorn's theorem (matrix theory) Specht's theorem (matrix theory) Spectral theorem (linear algebra, functional analysis) Sylvester's determinant theorem (determinants) Sylvester's law of inertia (quadratic forms)
The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. 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 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.
Rank–nullity theorem – In linear algebra, relation between 3 dimensions (by rank and nullity) Structure theorem for finitely generated modules over a principal ideal domain – Statement in abstract algebra; Jordan normal form – Form of a matrix indicating its eigenvalues and their algebraic multiplicities