Proving Linearity Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ Prove that
$f$ is linear $\iff f(\alpha x+\beta y) = \alpha f(x) + \beta f(y)$ for all $x,y \in \mathbb{R}^n$ and all $\alpha , \beta \in \mathbb{R}$.
I know that this statement is one of the definitions of linearity but I don't know how to prove it. Does it have something to do with the corresponding $m$x$n$ matrix that exists for this function?

Seeing as the question posed is an "if and only if" situation, we need to use each fact to confirm the other. If we have some linear function $ f:\mathbb{R}^n →\mathbb{R}^m $, then...
$$ (1): f(x+y)=f(x) + f(y)  $$
$$(2): f( \alpha x)=\alpha f(x) $$
If we have $f(\alpha x +\beta y)$ and $f$ is linear, using the first property (1):
$$ f(\alpha x+\beta y)=f(\alpha x) + f(\beta y). $$
Then using the second property (2):
$$f(\alpha x) = \alpha f(x) \ \ \ and  \ \ \ f(\beta y)=\beta f(y), $$
$$ f(\alpha x) + f(\beta y) = \alpha f(x) + \beta f(y).$$
hence, 
$$f(\alpha x + \beta y) = \alpha f(x) + \beta f(y).$$
Therefore $f$ is linear $ → f(\alpha x + \beta y) = \alpha f(x) + \beta f(y). $ It seems a bit trivial, I'm not sure if it's sufficient or not.
I'm also not sure how to prove that $ f(\alpha x + \beta y) = \alpha f(x) + \beta f(y) \rightarrow $ $f$ is linear. 
