Is Q(u,u) ( also $ \langle u,Av \rangle $ )considered a scalar product? If yes, explain why it is by writing how the scalar product is defined.

Question

I don't know if the following is a good answer: Let $u,v$ be vectors in $\Bbb R^n$, and let $A$ be a random $n×n$ matrix. Given that "$Av$" is actually a vector which a matrix has been applied to, we can consider $Av$ as $v$ in new coordinates imposed by the matrix $A$, also $ \langle u,Av \rangle $ is a normal scalar product? Can you tell me if this is a good point or I'm completely wrong?