Let S: $\displaystyle \mathbb{R}^2 \to \mathbb{R}^2$ be the function defined by S(x, y) = (x - y, y) for all (x,y) $\displaystyle \varepsilon \mathbb{R}^2$

Show that S is a linear map

Now I know that a linear map preserves the shape and takes lines to lines and points to points, but how would you show that S is a linear map using a formula such as

$\displaystyle S(\alpha \mathbf{x} + \beta \mathbf{y})$?