[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 youâ€™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 be an n-dimensional vector space with inner product and let be an orthonormal basis. If is an orthogonal transformation of , then for any point , we have

for all . 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.