Let $H = \{(x_1,x_2,x_3,x_4) : x_1 +x_2 - x_3 - x_4 = 0\}$, and let $T : \mathbb R^4 \to \mathbb R^4$ be the transformation given by reflection across $H$. I need to find a matrix representation of $T$, but I have no idea how to do this! I'm sorry for the lack of effort here, but I don't know how to generalize finding reflections across lines in $\mathbb R^2$ to any other dimension, would appreciate some hints.

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    $\begingroup$ This might help: en.wikipedia.org/wiki/Householder_transformation $\endgroup$ – user84413 Sep 20 '15 at 17:43
  • $\begingroup$ What user84413 said! There is no need to specify the hyperplane using a unit vector though. Here $\vec{v}=(1,1,-1,-1)$ is the natural normal, and the formula $$T(\vec{x})=\vec{x}-2\frac{\vec{x}\cdot\vec{v}}{||\vec{v}||^2}\vec{v}$$ works. $\endgroup$ – Jyrki Lahtonen Sep 20 '15 at 20:55

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