For example, if we know and are linearly independent then, span .
I understand there are many ways and theorems to do this, but I have just been learning about the method where you find the coefficients of the basis using Gaussian elimination.
I get how to do the manipulation, but I don't understand how it actually shows that a set of vectors span a space?
If you are given that the set is independent, then the only thing remaining to prove is that they span the space. That is, you want to solve the equation is solvable for any v in the space. If you are dealing with an n dimensional space, that reduces to n equations in n unknowns and, no matter what the right side is, that has a solution as long as Gaussian elimination does not give a row of all 0s.