Proof : Let X be a random variable and a be any constant.

$$\begin{array}{rcl}V[aX]& =& E[{(aX-E[aX])}^2]\\ & =& E[{(aX-aE[X])}^2](\text{∵}E[aX]=aE[X])\\ & =& E[\{a(X-E[X]){\}}^2]\\ & =& E[a^2{(X-E[X])}^2]\\ & =& a^{2}E[{(X-E[X])}^2](\text{∵}E[a]=a)\\ & =& a^{2}V[X]\end{array}$$