Showing $(Tp)(x) = x^2p(x)$ is a linear map (transformation) Define a linear map function $T: \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ where $\mathcal{P}(\mathbb{R})$ is the set of all polynomials with real-valued coefficients. Now let $T$ belong to the set of all possible linear maps from $\mathcal{P}(\mathbb{R})$ to itself, i.e.: $T \in \mathcal{L}(\mathcal{P}(\mathbb{R}), \mathcal{P}(\mathbb{R}))$ whereby we define this to be,
$$(Tp)(x) = x^2p(x) \tag{1}$$
for $x \in \mathbb{R}$ and where $p : \mathbb{R} \to \mathbb{R}$ is a polynomial function.
To show that (1) is a linear map we must show that it follows additivity and homogeneity. I'm not quite sure how to show that (1) homogeneous. Thanks for any help!
Note: the above can be found in Axler's Linear Algebra Done Right on page 39. 

let $p(x)=a_0+a_1x+ a_2x^2+\cdot\cdot\cdot\cdot+a_nx^n$
Now let $\lambda\in \mathbb R$ then $\lambda p(x)=\lambda a_0+(\lambda a_1)x+(\lambda a_2)x^2+\cdot\cdot\cdot\cdot+(\lambda a_n)x^n$
Then $T(\lambda p(x))=x^2\lambda a_0+(\lambda a_1)x^3+(\lambda a_2)x^4+\cdot\cdot\cdot\cdot+(\lambda a_n)x^{n+2}=\lambda x^2(a_0+a_1x+ a_2x^2+\cdot\cdot\cdot\cdot+a_nx^n)=\lambda T(p(x))$
