Uniqueness of basis vectors Say I have 2 vectors $v_1$ and $v_2$ as basis of a subspace. Then is it true that $kv_1$ and $mv_2$ where $k$ and $m$ are real numbers, are also basis for that subspace? 

If $k,m \neq 0$, you can write: $$v = av_1 + bv_2 = \frac{a}{k}(kv_1)+\frac{b}{m}(mv_2),$$ for $v \in \Bbb Rv_1 + \Bbb Rv_2$. Since $a$ and $b$ are unique for $v$, then you got a unique linear combination of $kv_1$ and $mv_2$, so we're done.
