Let

Find all nonempty subsets of which are linearly independent.

First, all of the one element subsets are linearly independent.

Next, none of the three and four element subsets are linearly independent (any set of more than two vectors in is dependent in by Theorem 12.10).

So, it is left to check the pairs. We know from a previous exercise (Section 12.15, Exercise #6) that two vectors and in are linearly independent if and only if . So, then we have