Kjernelinjer

Kjernelinjer, or kernel lines, are one-dimensional subspaces contained in the kernel (null space) of a linear transformation. For a linear map T: V → W between vector spaces, the kernel ker(T) consists of all vectors v in V such that T(v) = 0. When ker(T) has dimension one, it is a line through the origin, consisting of all scalar multiples of a single nonzero vector. This line is what is referred to as a kernel line.

Example: Consider the linear map T: R^2 → R given by T(x, y) = x. The kernel is ker(T) = { (0, y) | y ∈ R }, which is a vertical line through the origin in the plane. In general, kernel lines describe the one-dimensional directions in which the input can vary without affecting the output.

Computation and properties: To find kernel lines, solve the homogeneous system Ax = 0 for a matrix A representing T. Row-reducing A yields expressions for the free variables, leading to x = t v, where t ∈ R and v is a nonzero vector. The set {t v | t ∈ R} is the kernel line. The dimension of ker(T) is called the nullity, and the rank-nullity theorem states that nullity plus rank equals the dimension of the domain.

Applications: Kernel lines help characterize solution spaces of homogeneous systems, describe invariant directions in dynamical systems, and provide geometric intuition for constraints in linear models. They are the one-dimensional building blocks of larger kernel subspaces.

See also: kernel, null space, linear transformation, rank-nullity theorem.