Let $( | )$ be the standard inner product on $\mathbb{R^2}$ , and let $T$ be the linear operator $T(x_1,x_2)=(-x_2,x_1)$. Now $T$ is 'rotation through $90°$' and has the property that $(\alpha |T\alpha)=0$ for all $\alpha$ in $\mathbb{R^2}$ . Find all inner products $[ | ]$ on $\mathbb{R^2}$ such that $[\alpha | T\alpha]=0$ for each $\alpha$.
My Attempt since $\alpha\in \mathbb{R^2}$. So let us assume $\alpha=(x,y)$ then
$(\alpha |T\alpha)=((x,y) | T(x,y))=-xy+xy=0$ where $x,y\in \mathbb{R}$. So I think we have as many inner product on $\mathbb{R^2}$ as positive real numbers in $\mathbb{R}$.
Is it correct? Any help or hint will be appreciated.
Hint: Every inner product over $\Bbb R^2$ can be written in the form $[\alpha|\beta] = (S\alpha|S\beta)$ for an invertible linear map $S:\Bbb R^2 \to \Bbb R^2$.