RangNullitätTheorem

The Rank-Nullity Theorem is a fundamental result in linear algebra that relates the dimensions of two key subspaces associated with a linear transformation. Let T: V → W be a linear transformation between finite-dimensional vector spaces over a field F. Then the dimension of the kernel (null space) of T plus the dimension of the image (range) of T equals the dimension of V: dim Ker(T) + dim Im(T) = dim V. The rank of T is defined as dim Im(T), and the nullity of T as dim Ker(T). Therefore, rank(T) + nullity(T) = dim V.

A common way to prove the theorem is by using a basis argument. Take a basis for Ker(T) and extend it to a basis for V. The images of the additional basis vectors under T are linearly independent and span Im(T), so dim Im(T) equals the number of added vectors. This yields dim V = dim Ker(T) + dim Im(T), i.e., the rank-nullity relationship.

The theorem has several important consequences. It explains the degrees of freedom in solving linear systems: for a linear map represented by an m×n matrix A, rank(A) + nullity(A) = n, so the number of free variables equals the nullity. It also characterizes injectivity and surjectivity: T is injective iff nullity(T) = 0, and T is surjective iff rank(T) = dim W. The theorem is a cornerstone of dimension theory in linear algebra and underpins many applications in mathematics and applied sciences.