I'm trying to learn algebra and once again I don’t know the answer to one of my algebra textbook exercises.
Determine the set $ M_{\sigma} = \{ v \in V: ||v||=1, \sigma\beta|_{U\times U}$ is positive definite for $ U=\Bbb Rv\} $.
The symmetric bilinear form $\beta$ is defined by $ M_{E} = \left( \begin{array}{rrr} 1 & 0 \\ 0 & -1 \\ \end{array}\right)$,
$E$ being the standard basis, $ ||v||$ the standard Euclidean norm and $ \sigma \in\{+,-\} $.
So what I'm actually supposed to do is find one-dimensional subspaces of $ \Bbb R^2 $, so that the matrix is positive definite for for $\beta$ and for $-\beta$. But how?