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Let $\mathcal{H}: \mathbf{x}^T\mathbf{w}+b=0$ be a hyperplane in the $n$-dimensional Euclidean space of column vectors. Is there a way of "rotating" the above hyperplane such that it coincides with the $(n-1)$-dimensional hyperplane $x_2x_3...x_n$ in order the dimensions $x_2,...x_n$ to take values in $(-\infty, +\infty)$ and $x_1\in(c,+\infty)$?

Does it happen using special orthogonal groups in $n$-dimensions (SO($n$))? How exactly does it happen? Which is the value of $c$ above? Is it a change of variable $\mathbf{y} = R\mathbf{x}$, where $R\in SO(n)$? I have studied about special orthogonal groups, but I cannot find the answer...

Any hint on that would be great! Thanks in advance!

nullgeppetto
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  • Is there a feasible way to make this question be given more attention? I find this problem interesting, I think that many other people do so, but what can I do? – nullgeppetto Nov 13 '13 at 12:09
  • Yes! Just give it some bounty for it! I don't have enough credits to give part of it for this but if you collect enough points, you can! – Cupitor Nov 23 '13 at 18:42
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    I am also dealing with a similar problem, therefore I also feel the same; but just imagine how vast is the space of possible problems and you would realise its not strange that people might not be on the same freq as you are. – Cupitor Nov 23 '13 at 19:07

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