$f(g(x))=g(2)x$, linear transformation I saw an exercise with this unfamiliar for me linear transformation and I would like some help to find its matrix.
$f:\mathbb{R}_{2}[x]\rightarrow\mathbb{R}_{2}[x]$, $f(g(x))=g(2)x$
Here's my thought:
So let $B=\left \{ u_{1},u_{2},u_{3}\right \}=\left \{ (x^2+x+1),(x),(1)\right \}$ be a basis of $\mathbb{R}_{2}[x]$.
Then:
$f(u_{1})=(2^2+2+1)x=7x$
$f(u_{2})=2x$
$f(u_{3})=1$
So the matrix of $f$ in respect of $B$ is
$\begin{pmatrix}
0 & 0 & 0\\
7 & 2 & 0\\
0 & 0 & 1
\end{pmatrix}$
Is this correct? Or anywhere near correct?
 A: I see in the comments that there's some confusion as to what the question is. Since we can't look at your sources, we can't say whether you've copied the question correctly or whether there was something different.
That being said, the question you presented is perfectly meaningful. And your solution is almost correct. The only mistake is in your calculation of $f(u_3)$, which instead must be
$$f(u_3)=u_3(2)x=1x=x,$$
so the matrix of $f$ with respect to $B$ is
$$\begin{pmatrix}
0 & 0 & 0\\
7 & 2 & 1\\
0 & 0 & 0
\end{pmatrix}.$$
Some commented that your choice of a basis is strange. I agree that it's strange, but it's not wrong: $B=\left\{x^2+x+1,x,1\right\}$ is a basis of $\mathbb{R}_{2}[x]$. It's an unusual choice; people typically choose the standard basis $S=\left\{x^2,x,1\right\}$ for this vector space. But if it was given to you or even if you chose yourself to use it, there's nothing actually wrong with it.
NOTE: Of course, if the question is different — for example, if it's supposed to be $f(g(x))=g(2x)$ instead — then the answer will be different too.
