How to arrive at coordinates using 2D rotation matrix multiplication. I'm having a little trouble understanding 2D rotation matrices. I apologize in advance, as I have probably missed something really obvious! (My mathematics isn't brilliant!)
Ok, so I have the following matrices (the brackets should be joined vertically):
[x'] = [cos(a) -sin(a)]  [x]
[y']   [sin(a)  cos(a)]  [y]

Now, once I've multiplied x and y by the columns I am left with a new matrix of 4 numbers. However, I am supposed to end up with the following formulae to get the new coordinates:
x' = x cos(a) - y sin(a)
y' = x sin(a) + y cos(a)

My question is, how do I get from the 4 numbers of my matrix to the 2 numbers which will be the result of the above formulae. What happened to the rest of the sines and cosines? I imagine some kind of cancelling out?
Please could you explain the process to get from the matrices to formulae. Thank you so much!
(If my figures aren't clear, then they are available here!)

If you multiply your matrix with the vector, you get the results immediately. There are no steps in between. This is how matrix multiplication works.
