Find finite spanning sets 
I need help starting with the following questions:
Let P3 be the vector space of real valued polynomials of degree ≤ 3. Find finite spanning
  sets for the following subspaces of P3.
(a) T = {p ∈ P3 | (0) = 0}
(b) U = {p ∈ P3 | p'(0) = 0}
(c) V = {p ∈ P3 | p(1) = p(2)}
I do understand how to tell if a vector spans but how do I find the spanning sets? 

 A: I'll tackle the third case:
A generic element of $P_3$ is $p(x)=ax^3+bx^2+cx+d$
If $p(1)=p(2)$ then $a+b+c+d=8a+4b+2c+d$
With this condition we can eliminate one of the variables by solving for it. I choose to solve for $c$.
$c=-7a-3b$
As long as this condition is met, we will have a third degree polynomial satisfying the property that $p(1)=p(2)$.
Substituting back into the original polynomial gives a function of the form:
$p(x)=ax^3+bx^2+(-7a-3b)x+d$
We can then group the $a$'s $b$'s and $d$'s together to get our spanning set:
$p(x) = a(x^3-7x) +b(x^2-3x)+d=span \left[ x^3-7x,x^2-3x,1\right] $
In general, if you want to find a particular basis for a subspace of any given vector space, you can follow this procedure:


*

*Write out a generic vector in the larger space.

*Substitute in the defining property or properties of the subspace you are dealing with to eliminate parameters.

*Group the terms of the result in terms of the remaining parameters, and the result will be a linear combination of your basis vectors.

A: Well a general polynomial in $P_3$ takes the form $f(x) = ax^2 + bx + c$. Now let's solve (a), we want to have a more explicit description of $P$. If $f \in T$ then
$$ f(x) = 0 \iff c = 0 $$
Therefore
$$ P = \{ax^2 + bx : a, b \in \mathbb R \} $$
Then by observation a spanning set for this is $\{x, x^2\}$. (b) and (c) can be solved similarly, by finding what the spaces $U$ and $V$ actually look like.
