Prove that a set of matrices is a linear space

Prove that the set of matrices
$$v:=\left\{ \begin{pmatrix} 2x-y+z & x-2y-2z \\ x+y-z & 3x+y+2z \end{pmatrix} \middle|\, x,y,z \in R\right\}$$

Is a linear space above $R$ and find it's base.

As far as I know that for the set to be a linear space it needs to be closed under vector addition and under scalar multiplication, am I right?
but still I'm having a bit trouble structuring the proof