How to extend a set to fit a basis

Question

If I have the set $\{(3, −2, 1, 3),(−1, 3, −3, 4),(3, 8, 7, 0)\}$, how can I extend this into a basis of $\mathbb{R}^4$? I have seen questions in the past asking to reduce a set in order to fit it into a basis, but I am unsure how to do the opposite. Any help with this issue would be greatly appreciated.

Answer 1

Make a $4\times4$ matrix whose first three rows are your vectors, and whose last row is $(a\ b\ c\ d)$. Compute the determinant of this matrix (by expansion along the bottom row. would be a good way). Then choose any $a,b,c,d$ that make this determinant nonzero.