# Change of Basis and Coordinate system question

My apologies in advance: I am not very proficient in the field of mathematics. If my question seems discombobulated, it probably is, and I would greatly appreciate your help cleaning it up! Alright, onward...

I have a few coordinate systems to coordinate. I'm using a camera that detects markers in space and returns their location and attitude (in a vector and rotation matrix). I'm using a machine that moves in a room along standard Cartesian axes that are pre-determined in the room. The camera has its own coordinate system according to which it is detecting the markers.

I have a matrix, $R$ that can take a vector with respect to the room coordinate system and rotate it so that it is now expressed relative to the camera coordinate system.

I take record the position of the markers on the machine at an instant ($T_o$), and would like to report how much the machine has rotated ($D$) with respect to the room's coordinate system given the instantaneous positions of the markers ($T_i$).

$D=T_i^{-1}T_o$

yields the rotation in the camera's coordinate system.

$D=R^{-1}T_i^{-1}T_oR$

seems to give the correct answer, but I don't understand it.

Please let me know the formula for $D$ or, if the formula above is correct, could you give a quick explanation for it?

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Please don't yell (all capitals is the on-line equivalent of yelling). In this site, we can use italics and boldface for emphasis, so you can type *greatly* or _greatly_ to produce "greatly"; and **greatly** to get greatly. –  Arturo Magidin Apr 13 '11 at 19:25
What is the question? –  Qiaochu Yuan Apr 13 '11 at 19:33
I have modified my question. –  Limited Atonement Apr 13 '11 at 19:45