  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.
