Sum squared errors normal

Question

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$).

---

1. If $s_1,s_2$ are real numbers minimizing the sum of seuqred errors (SSE); then why is:

$Y_3^2+...+Y_n^2=SSE$; where

• $Y_1=k_0(s_1-a)$
• $Y_2=k_1(s_1-a)+k_2(s_2-b)$

• and $(Y_1,..,Y_n) = \frac{1}{\sigma} U (X_1-a-bc_1,..,X_1-a-bc_n)$ where $U$ is a unitary (real) matrix.

  2. Furthermore, what would the joint-distribution function of $a$ and $b$ be?