Search results
Results From The WOW.Com Content Network
Kernel and image of a linear map L from V to W. The kernel of L is a linear subspace of the domain V. [3] [2] In the linear map :, two elements of V have the same image in W if and only if their difference lies in the kernel of L, that is, = () =.
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.
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism f : X → Y is the quotient of Y by the image of f. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient.
Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed, [14] and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.
An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map.
Kernel (linear algebra) or null space, a set of vectors mapped to the zero vector; Kernel (category theory), a generalization of the kernel of a homomorphism; Kernel (set theory), an equivalence relation: partition by image under a function; Difference kernel, a binary equalizer: the kernel of the difference of two functions
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).