Let $U$ be the subspace of $V = P_{2}$ spanned by $f(x) = x^2 +x+1$. Find a subspace $W$ of $V$ such that $V = U\oplus W$.
I am having trouble with starting this question. I know that for $V = U\oplus W$ to be true, it needs to satisfy: (i) $V = U + W$ and (ii) $U\cup W =\{0\}$.
I am struggling to put this into polynomial format. I attempted the problem and got a subset that could satisfy this could be $\{x + x^2, 1 + x^2, 1 + x\}$. Is this correct?
It may help you to just cast this in terms of vectors. Represent polynomials $ax^2+bx+c$ as $\begin{bmatrix}a\\b\\c\end{bmatrix}$. Then the given polynomial is $(1,1,1)$ and $V=\operatorname{span}\{(1,1,1)\}$. If you can find a basis $\{(1,1,1),v,w\}$ for $\mathbb{R}^3$, then $\operatorname{span}\{v,w\}$ is one such subspace $W$.