If a row vector is multiplied by the 2D rotation matrix (therefore the "to be transformed" vector has to be infront of the transformation matrix)- must that 2D rotation matrix become transposed if the same vector is represented as a column vector and lies after the transformation matrix? I think that's how it goes, but most sites on the net state the 2D rotation transformation is cos() -sin() sin() cos() and hence assumes the vector is a column vector that lies after the transformation matrix.

Quote:Original post by Metro_Mystery
If a row vector is multiplied by the 2D rotation matrix (therefore the "to be transformed" vector has to be infront of the transformation matrix)- must that 2D rotation matrix become transposed if the same vector is represented as a column vector and lies after the transformation matrix?

Yes. Let's say you have the equation xA = y, and you want to switch to column vectors. You'd need to transpose 'y', which means you also need to transpose the other side of the equation for them to remain equal. You get (xA)T = yT, which equals ATxT = yT. Reversing the multiplication order when distributing a transpose or inverse operator is often overlooked.

Great answer- thanks champ!

I should also note that if you didn't change the multiplication order and just went from row vectors to column vectors, that the multiplication wouldn't have worked anyway (a 3x1 post-multiplied by a 3x3 is illegal).