Your question is not clear because you have not said what the dimemsion of your space is. Also when you say "the matrix containing the vectors", containing them how? As columns is most common but you should say that.

If, for example, you have two vectors in your set and it is in three dimensions, your "augmented matrix" will be three by three, not 4 by 4. I think it would be better to think about what "span" means, which you say you know. A vector, x, is in the "span" of the set of vectors {u, v, w}, if and only if there exist numbers a, b, c such that x= au+ bv+ cw. Of course, it you write out the components, you will have a system of equations you could then put into an "augmented matrix" which is essentially what you are doing. Saying that you get a 4 by 4 matrix implies that you have a set of 3 vectors each having 4 components. Is that correct?