Transformations with matrices are indeed very unintuitive and super fast. I am currently attending an introductory course about autonomous robotics and to keep the matter short in the end we "should" know all basics how to move a robot arm around involving forward and inverse kinematics.

I already learned a lot about transformations and it is actually for a computer super easy to calculate things and vector stuff when you do it right. I can especially recommend chapter 1 of the book "Introduction to Robotics, Mechanics and Control". Although it is focused on applying all the math on robotics but essentially it is all the same for the day to day vector calculations you need in a game. Seriously. Read and understand it - then you become a matrix guru in a vec_set-World ^^

Just to throw that in... it seems we are going off-topic.