In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: Properties The kernel of is a linear subspace of the domain . In the linear map two elements of have the same image in if and only if their difference lies in the kernel of , that is, From this, it follows that the image of is isomorphic to the quotient of by the kernel: In the case where is finite-dimensional, this implies the rank–nullity theorem: where the term refers the dimension of the image of , while refers to the dimension of the kernel of , That is, so that the rank–nullity theorem can be restated as When is an inner product space, the quotient can be identified with the orthogonal complement in 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 , 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