I have to find a basis for the subspace defined by
$$U=(p(x): p(1)=0)$$
of $P_2$ and then I have to find the Dim $U$
My steps so far have been solving for $U$ which resulted in $$(x^2-x,x^2-1)$$ which are both polynomials of degree 2 and that evaluate at $p(1)=0$
To check if something is a basis for another we have to check to see if the set of vectors are linearly independent and if $U=span(x^2-x,x^2-1)$
running $(x^2-x,x^2-1)$ through a linear independence test where I did,
$$0=s_1(x^2-x) +s_2(x^2-1)$$
which resulted in $s_1 = s_2 = 0$ for the above to hold true so the two vectors are linearly independent. Next I have to see if $U=(p(x): p(1)=0)$ which I have no idea how to do.
This is all I have so far and I am not sure if I even started the question correctly. If someone could lend me a guiding hand I would greatly appreciate it.
Note that the dimension of $\mathcal{P}_2$ is $3$ with the canonical basis $\{1,x,x^2\}$s. Now the $U\subset\mathcal{P}_2$, following the single constraint $p(1)=0$ for all $p\in U$. Hence the dimension of $U$ is at most $3-1=2$. But you have already found out two linearly independent vectors in $U$, which ten must form a basis of $U$.