Proving A Function Is Linear

Example

Consider $T:\mathbb{R}\to \mathbb{R},T(x)=7x$ Prove $T$ is a Linear Transformation

T Preserves addition

Pick $x,y\in \mathbb{R}$ $T(x+y)=7(x+y)$ defn of $T$ $=7x+7y$ as $\mathbb{R}$ is a field $=T(x)+T(y)$ by defn of $T$

T preserves Scaling

$T(cx)=7(cx)$ by defn of $T$ $=c(7x)$ as $\mathbb{R}$ is a field $=cT(x)$ by defn of $T$