Let $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a linear transformation be such that $\langle Tx,x\rangle =0$ for all $x\in \mathbb{R}^n$.
Then,
$\mathrm{trace}(T)=0$
$\det(T)=0$
all eigenvalues of $T$ are real
$T=0$
Well, if $x=\sum_{1}^{n}a_ie_i$ then the conditions implies that $\langle\sum_{1}^{n}a_iT(e_i),\sum_{1}^{n}a_ie_i\rangle=0$ but how to proceed next? please help.I mean which are correct and which are false?thank you for help.