Example of $\langle T(v),v \rangle =0$ for every $v$, but $T \neq 0$

Question

As it says in the title, I neeed an example of a linear transformation $T$, over $\mathbb{R}$ (not $\mathbb{C}$!), that satisfies $\langle T(v),v \rangle =0$ for every $v$ in $V$, but $T \neq 0$.

Any suggestions ? Thanks :)

Answer 1

There is no such transformation if $\dim V=1$.

An example for $\dim V=2$ is given by Marcin Łoś: $T\colon (x,y)\mapsto (-y, x)$

If $\dim V>2$, take orthogonal projection $P$ onto some two-dimensional plane. Then compose it with two-dimensional $T$.

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As an aside: if $\dim V$ is even, such $T$ may be constructed so that it's invertible. If $\dim V$ is odd, any such $T$ has nontrivial kernel.