What does it mean to say two real symmetric bilinear forms $A$ and $B$ on a vector space $V$ are congruent ?

State and prove Sylvester's law of inertia, and deduce that the rank and signature determine the congruence class of a real symmetric bilinear form. [You may use without proof a result on diagonalisability of real symmetric matrices, provided it is clearly stated.]

How many congruence classes of symmetric bilinear forms on a real $n$-dimensional vector space are there? Such a form $\psi$ defines a family of subsets $\left\{x \in \mathbb{R}^{n} \mid \psi(x, x)=t\right\}$, for $t \in \mathbb{R}$. For how many of the congruence classes are these associated subsets all bounded subsets of $\mathbb{R}^{n}$ ? Is the quadric surface

$$\left\{3 x^{2}+6 y^{2}+5 z^{2}+4 x y+2 x z+8 y z=1\right\}$$

a bounded or unbounded subset of $\mathbb{R}^{3}$ ? Justify your answers.