linear subspace of V. Thus, it makes sense to speak of the quotient space V/(ker T). The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image of T (which is a subspace of W). As a consequence, the dimension of V equals the dimension of the kernel plus the dimension of the image. If V and W are finite-dimensional and bases have been chosen, then T can be described by a matrix M, and the kernel can be computed by solving the homogeneous system of linear equations . In this case, the kernel of T may be identified to the kernel of the matrix M, also called "null space" of M. The dimension of the null space, called the nullity of M, is given by the number of columns of M minus the rank of M, as a consequence of the rank–nullity theorem. Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators. For instance, in order to find all twice-differentiable functions f from the real line to itself such that let V be the space of all twice differentiable functions, let W be the space of all functions, and define a linear operator T from V to W by for f in V and x an arbitrary real number. Then all solutions to the differential equation are in ker T. One can define kernels for homomorphisms between modules over a ring in an analogous manner. This includes kernels for homomorphisms between abelian groups as a special case. This example captures the essence of kernels in general abelian categories; see Kernel (category theory). Group homomorphisms Let G and H be groups and let f be a group homomorphism from G to H. If eH is the identity element of H, then the kernel of f is the preimage of the singleton set {eH}; that is, the subset of G consisting of all those elements of G that are mapped by f to the element eH. The kernel is usually denoted (or a variation). In symbols: Since a group homomorphism preserves identity elements, the identity element eG of G must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {eG}. If f were not injective, then the non-injective elements can form a distinct element of its kernel: there