Transformatiebasis

In linear algebra, transformatiebasis refers to a basis that is obtained by applying a linear transformation to an existing basis of a vector space. If V is an n-dimensional vector space over a field F and B = {v1, ..., vn} is a basis, and T: V → V is a linear transformation, then the set T(B) = {T(v1), ..., T(vn)} is called a transformatiebasis of V with respect to T.

If T is invertible, then T(B) is again a basis of V, because T preserves linear independence and spans V. If T is not invertible, T(B) spans the image im(T) and may not be a basis of V, although it still describes how the original basis vectors are transformed. The most familiar case is when V = R^n and the transformation is represented by a matrix A. The transformatiebasis is then the set of columns of A (when A is viewed as acting on the standard basis), and these vectors form a basis if A is invertible.

Transformatiebasis is closely related to change of basis. If B is the original basis and B' = T(B) is the transformatiebasis, the change-of-basis matrix from B to B' encodes how coordinates transform between the two bases. In computations, selecting an appropriate transformatiebasis can simplify the representation of T, such as by diagonalizing the operator when possible, or by aligning with particular geometric or computational goals.