Let $V$ be a finite-dimensional $\mathbb{R}$-vector space and $\beta$ a symmetrical Bilinear form on $V$. $\beta(v,v) \neq 0$ for $v \neq 0$. Also there is a certain $v_0 \in V$ , such that $\beta(v_0,v_0)=2022$.
Now I have to show that this Bilinear form is positive-definite.
I know that positive-definite means that $\langle v,v \rangle > 0$. Now this is true for this certain $v_0$ because $\langle v_0,v_0 \rangle=2022 > 0$ , but how would I show that this is true for all $v \in V , v \neq 0$. Is there a property that I'm missing that leads to the answer?
Best Answer
Since $V$ is finite-dimensional, you can use the Gram-Schmidt algorithm to construct an orthonormal basis $\{v_1,\ldots,v_n\}$ with respect to $V$. Then, assuming that any $v\in V$ satisfies $\beta(v,v)<0$, there must be some $v_i$ for which $\beta(v_i,v_i)<0$. See if you can work it out from there.