Finding a subspace such that a bilinear form is an inner product.

Question

Let

$$M = \left( \begin{array}{ccc} 2 & 2 & 2 \\ 2 & 5 & -1 \\ 2 & -1 & 5 \end{array} \right) $$

And $b: R^3 \times R^3 \rightarrow R$ be the bilinear form associated with $M$. Find a subspace $V \in R^3$ s.t. $b_{|V \times V}$ is a scalar product in $V$.

Where a scalar product is defined as being a bilinear form that is positively defined.

Is the solution to this simply $V = \{ (x,y,z) \in R^3 | 2x^2 + 5y^2 + 5z^2 + 4xy+4zx-2yz > 0 \}$ ?

Answer 1

If the question is just "find a subspace" (rather than the largest possible such space), we can take the set $$ \{(t,0,0):t \in \Bbb R\} $$