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# Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map L : V → W between two vector spaces V and W, is the set of all elements v of V for which L(v) = 0, where 0 denotes the zero vector in W. That is, In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map L : V → W between two vector spaces V and W, is the set of all elements v of V for which L(v) = 0, where 0 denotes the zero vector in W. That is, The kernel of L is a linear subspace of the domain V.In the linear map L : V → W, two elements of V have the same image in W if and only if their difference lies in the kernel of L: It follows that the image of L is isomorphic to the quotient of V by the kernel: This implies the rank–nullity theorem: where, by rank we mean the dimension of the image of L, and by nullity that of the kernel of L. When V is an inner product space, the quotient V / ker(L) can be identified with the orthogonal complement in V of ker(L). This is the generalization to linear operators of the row space, or coimage, of a matrix. The notion of kernel applies to the homomorphisms of modules, the latter being a generalization of the vector space over a field to that over a ring.The domain of the mapping is a module, and the kernel constitutes a 'submodule'. Here, the concepts of rank and nullity do not necessarily apply. If V and W are topological vector spaces (and W is finite-dimensional) then a linear operator L: V → W is continuous if and only if the kernel of L is a closed subspace of V. Consider a linear map represented as a m × n matrix A with coefficients in a field K (typically the field of the real numbers or of the complex numbers) and operating on column vectors x with n components over K.The kernel of this linear map is the set of solutions to the equation A x = 0, where 0 is understood as the zero vector. The dimension of the kernel of A is called the nullity of A. In set-builder notation,

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