Let $P^4$ be the vector space of polynomials of degree <4. Consider the following mapping : $F: P^4 \to P^4 $
$f(x) \to xf'(x) + f(x)$
Prove that this is a linear transformation:
My attempt: $f(x+y)=xf'(x+y) + yf'(x+y) + f(x+y)$ But now I'm a bit stuck, because if we substitute f(x+y) in the image, we have to do it again and again. Also, I don't see what I can do with the two $f'(x+y)$, how can I deal with them ? Thanks for your help.
You make a ( common ) mistake. The application you consider is $F : P^4 \rightarrow P^4$, $f \mapsto F\left(f\right)$ and you in your case have considered the function $x \mapsto F\left(f\right)\left(x\right)$.
Here for two polynomials $f$ and $g$ in $P^4$ and a scalar $\lambda$ $$ F\left(\lambda f + g\right)(x)=x\left(\lambda f + g\right)'\left(x\right)+\left(\lambda f + g\right)\left(x\right)=x\left(\lambda f'\left(x\right)+g'\left(x\right)\right)+\left(\lambda f(x) + g(x)\right) $$