Dimensionssatz

Dimensionssatz, in German mathematics, refers to the dimension theorem in linear algebra. It states that for a finite-dimensional vector space V over a field F and a subspace U ≤ V, the dimension of V equals the sum of the dimensions of U and the quotient space V/U: dim(V) = dim(U) + dim(V/U). Equivalently, dim(V/U) = dim(V) − dim(U). The quotient space V/U consists of the cosets v + U and captures the degrees of freedom of V not contained in U.

Conceptually, the theorem follows from choosing a basis of U and extending it to a basis of V. The extra basis vectors, when projected to the quotient, form a basis for V/U. This provides a concrete method for computing dimensions of quotient spaces and clarifies how subspaces contribute to the overall dimension.

The Dimensionssatz is closely related to other fundamental results. It underpins the rank-nullity theorem: for a linear map T: V → W, dim(V) = dim(Ker(T)) + dim(Im(T)). It also expresses that every short exact sequence 0 → U → V → V/U → 0 splits dimensionally in the finite-dimensional case.

Example: in V = R^3 with U = span{(1,0,0), (0,1,0)}, we have dim(V) = 3, dim(U) = 2, hence dim(V/U) = 1.

Variants and generalizations exist for infinite-dimensional spaces (where dimensions are cardinalities) and for modules over rings, where the straightforward sum formula may fail. The term is commonly called the dimension theorem in English, and Dimensionssatz is its standard German designation.