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Kernel (linear algebra) In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. [1] That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the ...
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 ...
The dimension of the column space is called the rank of the matrix and is at most min (m, n). [1] A definition for matrices over a ring is also possible. The row space is defined similarly. The row space and the column space of a matrix A are sometimes denoted as C(AT) and C(A) respectively.
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the ...
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1][2][3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. [4] Rank is thus a measure of the "nondegenerateness ...
Orthogonal complement. In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace of a vector space equipped with a bilinear form is the set of all vectors in that are orthogonal to every vector in . Informally, it is called the perp, short for perpendicular complement.
e. In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. [1] A graphical illustration of a zero-dimensional space is a point. [2]
Linear subspace. In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace[1][note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.