I'd like to discuss the following problem :
Write U = { $f \in V | \hspace{0.3 cm}x^{2} | f$},where | means "divides", and $ V = \mathbb{R}_{k}[x]$
As kernel of the linear application : $$ F : \mathbb{R}_{k}[x] \longmapsto \mathbb{R}^{s}$$
With k that doens't depend on s,
And write down the matrix of change basis in the canonical basis of $\mathbb{R}_{k}[x]$ and $\mathbb{R}^{s}$.
The second part should be quite easier once found what $F$ does.
My attempt was to pass in coordinates because the kernel of $ F : \mathbb{R}_{k}[x] \longmapsto \mathbb{R}^{s}$ should be equal to the kernel of $ \phi : \mathbb{R}^{k+1} \longmapsto \mathbb{R}^{s}$,
Trying to find $U$ in $\mathbb{R}^{k+1}$ writing $$\mathbb{R}^{k+1} = Span \begin{pmatrix}a \\ b \\ c+1 \\ \cdots \\ k \end{pmatrix} \bigoplus Span \{e_{1}, \cdots e_{s} \}$$ with $e_{3}$ missing, and $\{a,b,\cdots,k\} \in \mathbb{R}$,
And defining $\phi$ to be zero on the vectors of $Span \begin{pmatrix}a \\ b \\ c+1 \\ \cdots \\ k \end{pmatrix}$,
This will work ?
Any help would be appreciated, thank you all!
Observe that $U$ consists of all polynomials of the form $f(x)=a_kx^k+a_{k-1}x^{k-1}+...+a_2x^2$.
Now let $s=2$ and define $F : \mathbb{R}_{k}[x] \longmapsto \mathbb{R}^{2}$ as follows:
If $f \in V$ and $f(x)=a_kx^k+a_{k-1}x^{k-1}+...+a_2x^2+a_1x+a_0$,
then $F(f):=(a_1,a_0)$.
Show that $F$ has the desired properties.