Counterexample to linear transformation.

Question

What counterexample can I use to prove that ($ \mathbb{R}_{[x]}$is any polynomial):

$L :\mathbb{R}_{[x]}\rightarrow\mathbb{R}_{[x]},(L(p))(x)=p(x)p'(x)$ is not linear transformation. I have already proven this using definition but it is hard to think about example. I would be grateful for any help.

Answer 1

Let $p$ be given by $p(x)=x$ and let $q=2p$. We note that $L(p)(x)=x$ but $L(q)(x)=2x\cdot 2=4x$ But if $L$ was linear then $L(q)(x)=L(2p)(x)=2L(p)(x)=2x\neq 4x$ and so we reach a contradiction. Hence $L$ is not linear.

Answer 2

How about this? $$ L(x + x^2) = (x+x^2)(1 + 2x) = \dots $$ $$ L(x) + L(x^2) = \dots $$

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As a side note, if you want to prove that $L$ is not linear, you just have to provide one example where one of the properties fail. You say that you have proved it "using the definition" (I am not sure what you mean here), but this is not necessary. By providing a counter example, you have shown that the axioms don't hold.

Answer 3

$L(x+1)=(x+1)\cdot1$, whereas $L(x)+L(1)= x\cdot1+1\cdot0$.

Edit: evidently $L(2x)\neq2L(x)$.