# Thread: Showing that a set of linearly independent vectors are a basis?

1. ## Showing that a set of linearly independent vectors are a basis?

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?

2. For example, if we know $\displaystyle \dim E=n$ and $\displaystyle \{v_1,\ldots,v_n\}\subset E$ are linearly independent then, $\displaystyle \{v_1,\ldots,v_n\}$ span $\displaystyle E$ .

3. 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 $\displaystyle a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n= v$ 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.