About symmetric bilinear forms

Question

I am confused about symmetric bilinear forms. Here is my question: Let b is a positive definite symmetric bilinear form on real vector space. Is b non-degenerate?

Answer 1

Yes. To say that $b$ is non-degenerate is the same as to say that there is no $x$ such that the function $f_x: V \to V$ given by $$ y \mapsto b(x, y) $$ is identically the zero function.

To say that $b$ is positive definite means that for any $x$, $b(x, x) > 0$. In particular, $f_x(x) = b(x, x) \neq 0$ and so $f_x$ is not identically the zero function.