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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,

  1. $\mathrm{trace}(T)=0$

  2. $\det(T)=0$

  3. all eigenvalues of $T$ are real

  4. $T=0$

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.

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    $\LaTeX$ notes: do not use `<` and `>` for inner product, use `\langle` and `\rangle`. And don't use `\ni` for "such that". That symbol isn't the (bastardized) version of the "such that" symbol that one sometimes sees; it means "has `` as an element" That is, $X\ni a$ means "$a$ is an element of $X$".2012-07-08
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    Patience, you are flooding the site with questions. Why not slow down a bit and digest some of the answers?2012-07-08
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    Parts of it are true, but parts are not. Consider a rotation of $90^{\circ}$ on $\mathbb{R}^2$. Then $\langle Tx,x\rangle = 0$ for all $x$, but the characteristic polynomial of $T$ has nonreal roots.2012-07-08
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    Thank you, I must do that Gerry and Arturo.2012-07-08

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