Prove or disprove: quadratic form $Q(v):=\beta(v,v) $ (biliniarform) is nondegenerate if $\beta$ is.

Question

Let char$K \not = 2$ and $\beta$ be a bilinearform (not necessarily symmetric) on a $K$-vectorspace $V$. Let $Q$ be defined by ($v, w \in V$ arbitrary)$$Q(v) = \beta (v,v)$$ I've shown that $Q$ is a quadratic form with $$\beta _Q(v,w) = 0,5 \: \cdot \: (\beta(v,w)+\beta(w,v))$$ Now i have to prove or disprove the following:

$$\beta \textrm{ nondegenerate} \Rightarrow Q \textrm{ nondegenerate}$$ By using the definitions this boils down to showing that $\beta _Q$ is nondegenerate but i don't know how to show this.

Any ideas? (is my $\beta _Q$ maybe already wrong or is the statement simply false, so that i have to find a counterexample)

Thanks in advance!

Answer 1

I think if $\beta$ is a bilinear form nondegenerate $\Rightarrow$ $\beta(v,w)=0 \Leftrightarrow v $ or $w$ is $0$. So $Q(v)=0 \Leftrightarrow v=0$