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In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part (or linear subspace) of the domain which the map maps to the zero vector.[1] That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W,[2] or more symbolically:
Properties
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,
From this, it follows that the image of L is isomorphic to the quotient of V by the kernel:
When V is an inner product space, the quotient can be identified with the orthogonal complement in V of This is the generalization to linear operators of the row space, or coimage, of a matrix.
Application to modules
The notion of kernel also makes sense for homomorphisms of modules, which are generalizations of vector spaces where the scalars are elements of a ring, rather than a field. The domain of the mapping is a module, with the kernel constituting a submodule. Here, the concepts of rank and nullity do not necessarily apply.
In functional analysis
If V and W are topological vector spaces such that 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.
Representation as matrix multiplication
Consider a linear map represented as a m × n matrix A with coefficients in a field K (typically or ), that is operating on column vectors x with n components over K. The kernel of this linear map is the set of solutions to the equation Ax = 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,
The matrix equation is equivalent to a homogeneous system of linear equations:
Thus the kernel of A is the same as the solution set to the above homogeneous equations.
Subspace properties
The kernel of a m × n matrix A over a field K is a linear subspace of Kn. That is, the kernel of A, the set Null(A), has the following three properties:
- Null(A) always contains the zero vector, since A0 = 0.
- If x ∈ Null(A) and y ∈ Null(A), then x + y ∈ Null(A). This follows from the distributivity of matrix multiplication over addition.
- If x ∈ Null(A) and c is a scalar c ∈ K, then cx ∈ Null(A), since A(cx) = c(Ax) = c0 = 0.
The row space of a matrix
The product Ax can be written in terms of the dot product of vectors as follows:
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