# Math Help - Proving a function is a linear map

1. ## Proving a function is a linear map

Is the function $T : \mathbb{P}_3(\mathbb{R})\rightarrow \mathbb{P}_3(\mathbb{R})$ defined by $T(p)=q$ where

$q(x)=4xp'(x)-8p(x)$ for $x\in\mathbb{R}$

Would I set my proof as follows

Let $p, q\in \mathbb{P}_3(\mathbb{R}), \lambda\in\mathbb{R}$

$q(x)=4xp'(x)-8p(x), p(x)=4xq'(x)-8q(x)$

$T(p+q)=4x(p+q)'(x)-8(p+q)(x)$

$=4x(p'(x)+q'(x))-8(p(x)+q(x))$

$=4xp'(x)+4xq'(x)-8p(x)-8q(x)$

$=(4xp'(x)-8p(x))+(4xq'(x)-8q(x))$

$=T(p)+T(q)\Rightarrow \textrm{Addition condition satisfied}$

$T(\lambda p)=4x(\lambda q)'(x)-8(\lambda q)(x)$

$=4x\lambda q'(x)-8\lambda q(x)$

$=\lambda (4xq'(x)-8q(x))$

$=\lambda T(p)\Rightarrow \textrm{Scalar multiplication condition satisfied}$

Therefore, T is a linear map

2. Seems right to me.