a) u and v are nonsingular but is NOT nonsingular?
b) u and v are singular but is NOT singular?
Q2. Due to formatting ambiguities I can't tell what the two given matrices are.
Q3. What's P2? The set of second degree polynomials? If so, check whether:
1. B spans (that is, at^2 + bt + c can be constructed from a linear combination of the elements of B).
2. The elements of B are linearly independent.
For the second part, it's not clear to me what you're trying to find.