Did I find the basis of a subset of $\mathscr{P}_4(\mathbb{F})$?

Question

I need to find the basis of a subspace of $\mathscr{P}_4(\mathbb{F})$, called $U$, where $U=\{p\epsilon\mathscr{P}_4(\mathbb{F})|\ p''(6)=0\}$. After finding this basis I need to find another set, $W$, such that $\mathscr{P}_4(\mathbb{F}) = U\oplus W$.

I've reasoned that the basis of $U$ should be $\{1,\ x,\ (x-6)^3,\ (x-6)^4\}$, and $W$ should be $\{x^2\}$.

Is this a valid solution?

Answer 1

You have given a correct basis. One needs to give justification. This involves the following steps: (i) the proposed basis elements satisfy the condition on the second derivative, and therefore (ii) all linear combinations of them satisfy the condition.

Also (iii) the proposed basis is a linearly independent set.

Thus the subspace generated by the proposed basis is a subspace of $U$ and has dimension $4$. However, $U$ itself is a proper subspace of the full space, and therefore has dimension $\le 4$. It follows that the space generated by the proposed basis is all of $U$.

Your $W$ is essentially correct, it should be the space generated by $x^2$.