Descartes’ headache: rotation of space is transpose to rotation of co-ordinate system

[I often speak of such-and-such depending upon the choice of co-ordinate system, and proceed to show this by rotating the space. The following explains why this is equivalent (via the transpose)].

You awaken in a two-dimensional landscape. In front of you are co-ordinate axes, fixed to the ground on a pivot. The landscape is featureless except for a tree and a well. You get bored, and fall asleep again. When you awaken, you notice that the relative arrangement of the co-ordinate axes to the tree and the well is different. What happened?

You realise that you’ll never truly know what happened while you were asleep. Either (a) someone rotated the co-ordinate axes 90 degrees clockwise, or (b) someone rotated the world 90 degrees anti-clockwise about the pivot point of the axes.

Rotation of the space is equivalent (via the transpose) to rotation of the co-ordinate system. In fact, this is true more generally for orthogonal transformations and has a familiar mathematical expression. Let V be an n-dimensional vector space with inner product \langle \cdot , \cdot \rangle and let e_i, i=1, .., n be an orthonormal basis. If X is an orthogonal transformation of V, then for any point v \in V, we have

    \[\displaystyle $\langle Xv, e_i \rangle = \langle v, X^T e_i \rangle\]

for all i=1, .. , n. That is, the co-ordinates of the transformed point with respect to the original basis are the co-ordinates of the original point with respect to the transpose-transformed basis.

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