Kernel (algebra)
Kernel of a group homomorphism
The kernel of a group homomorphism f : G -> H consists of all those elements of G which are sent by f to the identity element of H. In formulas:
- ker(f) = {x in G : f(x) = eH}.
The kernel forms a normal subgroup of G.
One of the isomorphism theorem states that the factor group G/ker(f) is isomorphic to the image of f, the isomorphism being induced by f itself. A slightly more general statement is the fundamental theorem on homomorphisms.
Kernel of a linear operator
If A : V -> W is a linear transformation between vector spaces, the kernel of A is defined as
- ker(A) = {x in V : Ax = 0}
The kernel is a subspace of the vector space V, and again the quotient space V/ker(A) is isomorphic to im(A), the image of A; in particular, we have for the dimensions:
- dim(ker(A)) = dim(V) - dim(im(A)).
If V and W are finite-dimensional and bases have been chosen, then A can be described by a matrix M, and the kernel can be computed by solving the homogenous system of linear equations Mx = 0. The dimension of the kernel is given by the number of columns of M minus the rank of M.
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 such that x f ''(x) + 3 f '(x) = f(x), one has to consider the kernel of the linear operator A : V -> W, where V is the vector space of all twice differentible functions, W is the vector space of all functions, and for f in V, we define Af(x) = x f ''(x) + 3 f '(x) - f(x).
One can define kernels for homomorphisms between modules in an analogous manner.
Kernel of a ring homomorphism
The kernel of a ring homomorphism f : R -> S consists of all those elements x of R for which f(x) = 0. Such a kernel is always an ideal of R.
The isomorphism theorems mentioned above for groups remain valid in the case of rings.
Universal algebra
All the above cases are unified and generalized universal algebra as follows: given algebras A and B of a certain type and a homomorphism f from A to B, the kernel of f is the congruence on A defined as follows: given elements x and y of A, let x ~ y iff f(x) = f(y). Then the relation ~ is the kernel of f.
In the case of groups, if f is a group homomorphism from G to H, the two notions of kernel are related as follows: given a and b in G, a ~ b iff f(a) = f(b), but that holds iff f(b)-1 * f(a) is the identity element eH of H. Since f is a homomorphism, this is true iff f(b-1a) is eH. So to know whether a ~ b, it's enough to keep track of the subgroup {x in G : f(x) = e} of G consisting of those elements of G that are mapped by f to the identity of H; it is this subgroup that we earlier called the kernel of fearlier. a ~ b iff b-1a belongs to that subgroup.
In more general universal algebra, kernels cannot be thought of as subalgebras but must be thought of as congruences.
Category theory
The notion of kernel of a morphism in category theory is a different generalization of the kernels from groups, rings and vector spaces.