Let be a column vector of independent N(0,1) distributed random variables. Let A be an orthogonal matrix.

(1) Consider the random column vector , obtained via the linear transformation . Show that is again a vector of independent N(0,1) distributed random variables.

(2) Let A be an orthogonal matrix whose last row is . Consider again the linear transformation . Show that

(3) Let be independent N random variables, with sample variance . Show using (1) - (2) that