I am just checking different analogous of $\alpha:V \longrightarrow W$ being affine. I have problems with this one:
$\alpha:V \longrightarrow W$ affine $\iff$ $G_\alpha=\{(v,\alpha(v): v\in V)\}$ is a affine subapce of $V\times W$.
Intuitively ($\Leftarrow$) should be clear, because any linear combination of $(v,\alpha(v))$, $\sum_{i=1}^{k}\lambda_i (v,\alpha(v))$ with $\sum_{i=1}^{k}\lambda_i=1$ is still in A but how is possible to formalize it in a nice way?
($\Rightarrow$) is not so clear to me