I'm having problems solving this exercise and would appreciate some help.
Let $(V, \varphi)$ be a non-degenerate quadratic space of dimension $\geq 2$. Show that there exists a subspace $P\subset V$ of $dim = 2$ such that the restricion of $\varphi$ to $P$, $\varphi_{P}$ is non degenerate.
I thought I could show this with the argument that if $\varphi$ is non-degenerate, then the determinant of the matrix associated to $\varphi$ has non zero determinant and thus the determinant of a minor $2\times 2$ matrix is also non zero.
Is my idea correct? Any help would be really appreciated. Thanks.
Not every single minor of a nonsingular matrix will be nonzero. You can prove this just applying the definition of non-degeneracy:
If $(V,\varphi)$ is non-degenerate then for each $v\in V$ there is a $w\in V$ with $\varphi(v,w)\neq 0$ (just check that you can take $w\not\in \text{span}(v)$). Now take $P=\text{span}(v,w)$ and check that the restricted quadratic space will also be non-degenerate.