kärnform

Kärnform, in Swedish mathematical terminology, refers to what is commonly called the kernel of a function or linear transformation. The kernel consists of all inputs that map to the zero element of the codomain.

For a linear map T: V → W, the kernel is ker(T) = {v in V | T(v) = 0}. It is always a subspace of V. The dimension of ker(T) is called the nullity; the rank-nullity theorem states that if V is finite-dimensional, dim(ker T) + dim(Im T) = dim V.

To determine the kernel of a matrix A, one solves the homogeneous system Ax = 0. Row-reduction yields a parametric description of ker(A) by expressing the solution in terms of free variables and then extracting a basis for the kernel.

Example: Let A be the 2×3 matrix [[1, 0, 0], [0, 1, 0]]. Solving Ax = 0 gives x1 = 0 and x2 = 0, while x3 is free. The kernel is the one-dimensional subspace spanned by (0, 0, 1).

Context and usage: The kernel or kärnform is central in solving linear systems, understanding linear relations among variables, and in areas such as differential equations and representation theory, where kernels identify invariant structures.

Etymology and usage: Kärnform is the Swedish term for kernel; in English-language texts the word kernel is standard. In Swedish education and mathematics literature, kärnform appears as the formal label for the kernel concept.