# Why EKF-SLAM uses EKF instead of linear Kalman Filter?

I am writing a bachelor's diploma on vSLAM. I learned and programmed EKF-SLAM like it's written in MonoSLAM paper, and I was going to write, that I can't use KF and have to use EKF because of non-linearity of observation function, but wait, how is it possible, if everything is linear?!

I understand, that if I store direction of a camera in form of axis-angle vector or quaternion then there will be something non-linear, but what if I will store it directly as values of rotation matrix? Then my observation function is just going to be a multiplication of matrices, which is a linear operator, and therefore linear. Am I wrong?

You can store them in your state space however that just increases the dimension of your state space since instead of three euler angles, you now have 6 terms (cosine and sine value of each angle). This is really undesirable in nature as it unnecessarily increases the computation time of the over all algorithm (which is of the order $O(n^{2.4})$ in EKF SLAM.