RangNullität

RangNullität, or the rank-nullity theorem, is a fundamental result in linear algebra that relates the dimensions of subspaces associated with a linear transformation. Given a linear map T: V → W between vector spaces over a field F, one defines the image (or range) Im(T) as the set of all outputs and the kernel (or null space) Ker(T) as the set of all vectors sent to zero. The rank of T is dim(Im(T)) and the nullity of T is dim(Ker(T)).

For finite-dimensional V and W, the Rank-Nullity Theorem states that dim(V) = dim(Ker(T)) + dim(Im(T)). Equivalently, dim(V) = nullity(T) + rank(T). In matrix form, for A ∈ F^{m×n}, rank(A) + nullity(A) = n, where nullity(A) is the dimension of the solution space of A x = 0.

Consequences include: T is injective if and only if Ker(T) = {0}, i.e., nullity is zero. T is surjective if and only if Im(T) = W when W is finite-dimensional, which means rank(T) = dim(W). For matrices, a rank deficiency corresponds to nontrivial solutions of Ax = 0, and the remaining degrees of freedom in the input give the kernel.

A brief proof outline: choose a basis of Ker(T), extend it to a basis of V, and then the images of the remaining basis vectors form a basis for Im(T). Counting dimensions yields dim(V) = dim(Ker(T)) + dim(Im(T)).

The theorem extends to infinite-dimensional spaces using cardinalities, where dim V = dim Ker(T) + dim Im(T) as cardinals; it is commonly referred to as the dimension theorem.