If $\{u,v,w\}$ is a spanning subset of $V$ then so is $\{u+v, v+w, w+u\}$ I've looked at the following solution but am unsure if it translates to this problem. Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.

That problem is very similar to this one- if you read the proof, you can see that it uses linear independence of the first basis to prove linear independence of the other set, and the fact that the first basis spans $V$ to show that the second set does. You can just mimic the style of proof in the linked question to solve this one. To start you off, let $x$ be a vector in $V$. Then for some scalars $a,b,c, au+bv+cw=x$ since $\{u,v,w\}$ spans $V$. Can you find some scalars $a',b',c'$ such that $a'(u+v)+b'(v+w)+c'(u+w)=x$?
