This was a bonus question on my test yesterday (I hope I can remember it correctly).

Q: Let W be a subsapce of $\displaystyle P_{3}$ such that P(0)=0 and let U be a subspace of $\displaystyle P_{3}$ such that P(1)=0. Find a basis for both W and U.

A: For W I figured the basis ought to be $\displaystyle \{x,x^{2},x^{3}\}$ since I don't wan't $\displaystyle a_{0}$ to have any value. But, I am stuck on U. I think I am seeing it all wrong. My x's are my vectors correct? As in, $\displaystyle \{1,x,x^{2},x^{3}\}$ is my basis for all third degree polynomails which have the form $\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$. I tried it a couple different ways on the exam, but should said they were all wrong, but that I was close. Even so, I think I am stuck in a rut and I am having a hard time starting from scratch. Im having trouble finding a systematic approach.

Thanks