This isn't false; it's a theorem. Using the axiom of choice, all vectors of any space can be represented as finite linear combinations. In the case of infinite-

This isn't false; it's a theorem. Using the axiom of choice, all vectors of any space can be represented as finite linear combinations. In the case of infinite-dimensional vector spaces, this means you would need at most n vectors from the infinite basis B, and n scalars from the underlying field F. I'm interested in what you're trying to demonstrate with your example though, or if we misunderstand each other, because this theorem is very well established.