I am on the following problem from a maths book:
Be $E$ the vector space of polynomials in $x$ with complex coefficients of degree strictly less than $n, n > 2$ and $x_1$, $x_2$ two distinct elements in $\mathbb{C}$.
a) Show that the following subsets of E are sub vector spaces of E: the set $F_1$ of polynomials with the root $x_1$, the set $F_2$ of polynomials with the root $x_2$, the set F of polynomials with roots $x_1$ and $x_2$. Find their dimensions.
b) Show that E is the direct sum of $F_1$ + $F_2$
I could solve easily part a) and find that the dimensions of $F_1$ and $F_2$ are $n - 1$ and dimension of $F$ is $n - 2$.
However, I am stuck on part b) that seems to be illogical since $\dim(E)$ is $n$ or do I have misunderstood something?
Thanks for your support
$F_1$ (or $F_2$) is a proper(!) subspace of $F_1+F_2$, hence $\dim(F_1+F_2)>\dim F_1=n-1$. This shows that $F_1+F_2=E$. However, contrary to what you wrote, $E$ is in general not the direct sum of $F_1$ and $F_2$.