In the post Can You See in Four Dimensions?, we saw some ways of visualising functions plotted in four-dimensional ‘space’ in various ways. Of course, we used colour and time for two dimensions because it is a bit difficult to plot in four actual spatial dimensions!

Here is another example: the orthogonal group $ \mathrm{O}_2(\mathbb{R})$ over the real numbers $ \mathbb{R}$. This is the group of all matrices $ A\in \mathrm{GL}_2(\mathbb{R})$ such that $ AA^t = I_2$, where $ I_2$ is the $ 2\times 2$ identity. Let

$ A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$

be such a matrix. Then $ a = \pm\sqrt{1 – b^2}$ and $ c = \pm\sqrt{1 – d^2}$. Finally, $ ac + bd =0$. As long as these conditions are satisfied, then $ A$ will indeed be in the orthogonal group.

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