Let $F$ be a field and let $V$ be the subspace of $F[X]$ consisting of all polynomials of degree at most $4$. Find a complement for $V$ in $F[X]$.

Question

Let $F$ be a field and let $V$ be the subspace of $F[X]$ consisting of all polynomials of degree at most $4$. Find a complement for $V$ in $F[X]$.

So I need to find a subspace $w$ such that the direct sum of $W$ and $V$ is $F[X]$. Any solutions or hints are greatly appreciated.

Answer 1

Hint:

For any polynomial $P(X)$, you can write: $$P(X)=a_0+a_1X+a_2X^2+a_3X^3+a_4X^4+X^5Q(X),\qquad Q(X)\in F[X].$$

Answer 2

Let $[v_0,\ldots,v_k]$ be a basis of$~V$ (which is finite dimensional), and let $[b_0,b_1,\ldots]$ be a basis of$~F[X]$ (which is countably infinite dimensional); I'm sure you can find such bases. Now put them together in on list $[v_0,\ldots,v_k,b_0,b_1,\ldots]$, which sequence of vectors spans all of $F[X]$ (since the $b_i$ already do), but is not linearly independent. To make it linearly independent, throw out any vector of the sequence that is in the span of the (retained) vectors that come before it in the sequence. Since $[v_0,\ldots,v_k]$ are linearly independent, none of them will be thrown out. As only redundant vectors were removed, all retained vectors still span $F[X]$, and being linearly independent they form a basis; it starts with $[v_0,\ldots,v_k,\ldots]$.

Now the set of vectors $b_i$ that were retained span a complementary subspace to the span of $[v_0,\ldots,v_k]$, that is, complementary to$~V$.