Let S and T be the subspaces of P3 consisting of all polynomials p(x) such that p(0)=0, and let T be a subspace of all polynomials q(x) such that q(1)=0. Find dim(S), dim(T), and dim(S intersect T).

Question

anonymous · Answer

i think S has as a basis $\{x, x^2\}$

anonymous · Answer

a three term polynomial $p$ with $p(0)=0$ looks like $ax^2+bx$

anonymous · Answer

this is a pretty easy check actually, because if $ax^2+bx=0x^2+0x$ then $a=b=0$ showing they are linearly independent

anonymous · Answer

oh here, a quick google search finds this
maybe it will help, since it is pretty much exactly your problem

anonymous · Answer

Yeah I saw that and that is helping me so far. I'm just not sure about the dimensions that I need to find.

anonymous · Answer

isn't the dimension of S 2 ?

anonymous · Answer

$P_3$ is spanned by $1,x,x^2$ and $S$ is spanned by $x,x^2$

anonymous · Answer

Yeah, wow I'm dumb. I knew that.

anonymous · Answer

i thought maybe there was something more complicated going on, but i think once you have the spanning vectors you have the dimension

anonymous · Answer

Yeah thanks. Guess I just needed someone to help me see it.

anonymous · Answer

yw

anonymous · Answer

Doesn't $P_3(x)$ mean all polynomials of degree at most 3?