I have a small confusion about functions and variables. So, on doing a bit of reading, a linear transformation is a function that maps inputs from one vector space to another. So, let us take for example a simple rotation matrix. This matrix takes a point in 2D space and maps it to another point in this 2D space. So, this is fine and the argument to this rotation function is a 2D point. Now, in many applications, we want to find the optimal rotation matrix i.e. we want to find the optimal angle [itex]\theta[/itex] to do the rotation. So, my question is can we view this rotation matrix now as a function of two variables i.e. [itex]\theta[/itex] and the input 2D point. All the explanations seem to treat the angle of rotation as a constant or in our optimisation problem case some unknown (but constant) quantity to estimate. So, in most optimisation cases it will involve taking the derivative of the rotation matrix wrt to the variable of interest i.e. [itex]\theta[/itex] and I was wondering whether I can view the rotation matrix as a function of [itex]\theta[/itex] i.e. write it as [itex]R(\theta, p)[/itex] where [itex]R[/itex] is the rotation function and [itex]p[/itex] is the point in 2D space.