Part 1 looks good to me, but for number two, your second basis element is not even in the set .
I think these are the ones you want.
Linear independence and spanning are both pretty clear.
1. Find three distinct non-zero vectors in such that but such that .
Consider , , and . Then . But .
2. Find a basis for , the vector space of matrices with trace zero. Why is this set a basis?
So we want to find a spanning set that is linearly independent. Consider where and can be chosen freely. This is a basis because it is a linearly independent set that spans .
Is this correct?