How can I prove the following lemma?

Let $mathbf{X}^ prime$ = $ left[ X_1 , X_2 , ldots, X_n right]$ where $ X_1, X_2, ldots X_n $ are observations of a random sample from a distribution which is $N left ( 0,sigma^2 right)$. Then let $mathbf{b}^prime = left[b_1,b_2,ldots,b_n right]$ be a real nonzero vector and let $mathbf{A}$ be a real symmetric matrix of order $n$. Then $mathbf{b ^prime X}$ and $ mathbf{X} ^prime mathbf{A} mathbf{X} $ are independent **iff** $mathbf{b} ^prime mathbf{A}=0.$

I know that $mathbf{b^ prime X } sim N(0, sigma^2 mathbf{ b^prime b})$ but I do not see how I could proceed here. My main difficulty lies on the fact that these two variables are very different; had it been two quadratic forms instead, then Craig's theorem would be of use.

Any advice?

Thank you.

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#### Best Answer

Use Craig's Theorem. Consider the quadratic form on **b**. If two random variables are independent, then any univariate functions of those random variables are likewise independent. The quadratic forms are independent, ergo the linear form on **b** and the quadratic form on **A** are likewise independent.

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