About symmetric bilinear forms 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?

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.
