A normal form for bases of finite-dimensional vector spaces

Most algorithms constructing bases of finite-dimensional vector spaces return basis vectors which, apart from orthogonality, do not show any special properties. While every basis is sufficient to define the vector space, not all bases are equally suited to unravel properties of the problem to be solved. In this paper a normal form for bases of finite-dimensional vector spaces is introduced which may prove very useful in the context of understanding the structure of the problem in which the basis appears in a step towards the solution. This normal form may be viewed as a new normal form for matrices of full column rank.