Extending a set to a basis

Question

How do I extend a set to a basis? Show that vectors v1 = (1, 2, 0) and v2 = (1, 3, 1) are linearly independent, then extend the set {v1, v2 } to a basis for R3. I understand how to do the first part of the question, it is the second part that I do not understand.

Answer 1

Just take any vector $u$ which is not in the plane determined by $v_1$ and $v_2$, that is you can't write $u$ as $av_1+bv_2$ for some scalars $a$ and $b$. Or you can take just $v_1\times v_2$. This one is perpendicular on the plane determined by $v_1$ and $v_2$.