Let $\displaystyle X = (X_1,...,X_n)^T$ be a column vector of independent N(0,1) distributed random variables. Let A be an $\displaystyle n \times n$ orthogonal matrix.

Prove that $\displaystyle ||X||^2 = X_1^2 + ... + X_n^2 \sim \mathcal{X}_n^2 $ using moment generating functions.