Prove that $\Var[\alpha X] = \alpha^2 \Var[X]$ from the definition (C.27) of variance.

Easy. We just use the linearity of expectation:

$$ \Var[\alpha X] = \E[\alpha^2 X^2] - \E^2[\alpha X] = \alpha^2 \E[X^2] - \alpha^2\E[X] = \alpha^2 (\E[X^2] - \E^2[X]) = \alpha^2 \Var[X] $$