Which of the following maps $T: V \rightarrow V$ are linear transformation?

Question

Let $V$ be the vector space of polynomials in $X$ with coefficient in $\mathbb{R}$. Which of the following maps $T: V \rightarrow V$ are linear transformation?

1) $T(p(X))=p(X^2)$ for all $p(X) \in V$

2) $T(p(X))=p(X)^2$ for all $p(X) \in V$

3) $T(p(X))=X^2p(X)$ for all $p(X) \in V$

4) $T(p(X))=p(X^2+1)$ for all $p(X) \in V$

My work: I find option 1, 3, 4 are satisfied the condition for being a linear a transformation, Is that correct?

Answer 1

yes, except 2 they are all linear transformations.

for 2 just consider the polynomial $p _1(x) = p _2(x)= x$

you can easily check that $T( p _1(x) + p _2(x)) \neq T( p _1(x) )+ T(p _2(x)) $

Answer 2

More generally: you can easily prove (try it!) that for any fixed polynomial $g$, $$C_g(p) = p\circ g$$ and $$P_g(p) = g\cdot p$$ are linear.