# Thread: Proving a function is a linear map

1. ## Proving a function is a linear map

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

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

Would I set my proof as follows

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

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

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

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

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

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

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

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

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

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

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

Therefore, T is a linear map

2. Seems right to me.