Change of Frame

It is sometimes useful to be able to tranform the affine coordinates of a point P in terms of some frame into its representation in terms of another frame. We can derive the formula for this change of frame operation.

Suppose P has the coordinates

relative to the frame

We wish to determine the coordinates of P relative to the frame

In other words, we wish to find (p1',...,pn') such that

Well, observe that each element of the new frame has some coordinates in terms of the old frame. We can express this relation with the set of equations

This has a natural matrix representation as

Once we have this matrix, it follows from above that we can compute P's coordinates with respect to the new frame by

Finally, we usually express this equation in column form:


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