I have a field $k$ and a finite dimensional $k$-vector space $E$. Let $f$ be a symmetric $k$- bilinear form on $E$. I define $f$ to be non-degenerate if $f(x,y)=0$ $\forall y\in E$ implies $x=0$. I define $f$ to be nonsingular if the map $E \to {\rm Hom}_k(E,k)$ defined by $x \mapsto \varphi_x$ where $\varphi_x(y)=f(x,y)$ is an isomorphism.
How can show that $f$ is non-singular iff it's non-degenerate?
I guess the 'if' part might be quite easy since $E$ is finite dimensional.
Hint: You can show (if you don't know already) that $\dim \operatorname{Hom}_k(E, k) = \dim E$. As such, a $k$-linear map $E \to \operatorname{Hom}_k(E, k)$ is an isomorphism if and only if it is injective.