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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.
In the case where V is finite-dimensional, this implies the rank–nullity theorem: () + () = (). where the term rank refers to the dimension of the image of L, (), while nullity refers to the dimension of the kernel of L, (). [4] That is, = () = (), so that the rank–nullity theorem can be ...
In this case, the kernel of T may be identified to the kernel of the matrix M, also called "null space" of M. The dimension of the null space, called the nullity of M, is given by the number of columns of M minus the rank of M, as a consequence of the rank–nullity theorem.
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 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).
Cayley–Hamilton theorem; Spread of a matrix; Jordan normal form; Weyr canonical form; ... Null space, nullity; Rank–nullity theorem; Nullity theorem; Dual space ...
By the rank-nullity theorem, dim ... Since the rank of a matrix is preserved by similarity ... (1985), Matrix Analysis, Cambridge University Press ...