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 $\{u+v, v+w, u+v+w\}$.

It is true. Subtract one of the three from another, and add it to the third one. You will get the original set.
