find a linearly independent subset F of E

csc2iffy

1. The problem statement, all variables and given/known data
Let U be the subspace of R5 spanned by the vectors
E={(1,1,0,0,1),(1,1,0,1,1),(0,1,1,1,1),(2,1,-1,0,1)}.
Find a linearly independent subset F of E with Span(E)=U


2. Relevant equations



3. The attempt at a solution
I figured out that E is linearly dependent and that the solution set to it is
c1=-t, c2=-t, c3=t, c4=t
I'm not sure how to figure out which vectors are linear combos of the each other??
I'm assuming that once I find the linearly dependent one(s), then the remaining vectors will be linearly independent. Will this be F?

csc2iffy

please please PLEASE help me someone, I have a final on this at 8AM :(

Mark44

From your work, the vectors in E are linearly dependent. If you set t = 1, you have -1v₁ + (-1)v₂ + 1v₃ + 1v₄ = 0, where the v_i vectors are the ones in your set.

This equation can be solved for v₄ as a linear combination of the other three, so you can discard v₄. If the resulting set is now linearly independent, then that's your answer. If the resulting set is still linearly dependent, keep discarding vectors until you get a set of vectors that is linearly independent.