What is a quadratic form on a finite dimensional real vector space $V$ ? What does it mean for two quadratic forms to be isomorphic (i.e. congruent)? State Sylvester's law of inertia and explain the definition of the quantities which appear in it. Find the signature of the quadratic form on $\mathbb{R}^{3}$ given by $q(\mathbf{v})=\mathbf{v}^{T} A \mathbf{v}$, where

$$A=\left(\begin{array}{ccc} -2 & 1 & 6 \\ 1 & -1 & -3 \\ 6 & -3 & 1 \end{array}\right)$$