Does $\mathrm{span}\{u, u-v\}$ contain the vector $v$? Am I thinking about this correctly?  
\begin{aligned}
\mathrm{span}\{u,u-v\} &=  c_1u+c_2(u-v)\\ &= c_1u + c_2u - c_2v \\ &= c_1u + c_2u + c_3v
\end{aligned}
where $c_3=-c_2$. So we have  $$\mathrm{span}\{u,u-v\}=(c_1+c_2)u + c_3v$$ which is just $\mathrm{span}\{u,v\}$ isn't it?  Am I making a huge conceptual mistake here?
Thanks in advance.

You have $$v=u-(u-v). $$ So $v $ is a linear combination of $u $ and $u-v $.
