Originally Posted by

**roninpro** You've identified the necessary items. We can start from there.

First, suppose that $\displaystyle S=\{v_1, v_2, w\}$ is linearly dependent. Then, we can find $\displaystyle c_1,c_2,c_3$ (not all zero) so that $\displaystyle c_1v_1+c_2v_2+c_3w=0$, and we may assume that $\displaystyle c_3\ne 0$. (Why?) Rearranging gives $\displaystyle w=\frac{c_1v_1+c_2v_2}{c_3}$. Any problems with this?

To show $\displaystyle \span S=V$, we can write $\displaystyle w=b_1v_1+b_2v_2+b_3v_3$ for some $\displaystyle b_1,b_2,b_3$, where, in particular, $\displaystyle b_3\ne 0$. Now, $\displaystyle S=\{v_1, v_2, b_1v_1+b_2v_2+b_3v_3\}$. Can you see how this helps?

Good luck.