Kerndimensie

Kerndimensie is a concept in linear algebra that refers to the dimension of the kernel of a linear transformation. For a linear map T: V → W, the kernel is the set ker(T) = {v in V | T(v) = 0}. The kerndimensie is the dimension of this set, and it is also called the nullity of T, often written nullity(T) or dim(ker(T)).

In finite-dimensional settings, the kernel is a subspace of V. The nullity measures how many degrees of freedom remain in solving the homogeneous system T(v) = 0. The rank of T, defined as dim(Im(T)) where Im(T) is the image of T, together with the nullity is governed by the Rank-Nullity Theorem: dim(V) = rank(T) + nullity(T). For matrices A, ker(A) is the solution space of Ax = 0, and nullity(A) = number of columns n minus rank(A).

Example: Let A be the 2×3 matrix [[1, 0, 0], [0, 1, 0]]. Its rank is 2 and n = 3, so nullity(A) = 1. The kernel consists of all multiples of the vector (0, 0, 1), hence dim(ker(A)) = 1.

The kerndimensie can also be defined for linear maps between arbitrary vector spaces, and in infinite-dimensional contexts its value may be finite or infinite. In such cases, the Rank-Nullity relationship is expressed in terms of appropriate notions of dimension or cardinality. In short, kerndimensie quantifies the amount of nontrivial solutions to T(v) = 0 and plays a central role in understanding the structure of linear maps.