I am trying to learn tensor calculus, but I must be confused about tensor invariance. I know the definition of a tensor is a number or function that transforms according to certain rules under a change of coordinates. The transformation leaves the number or function invariant if it is a tensor. Here is where I am confused-- when they speak of change of coordinates. For example: Let's say there is a vector in an orthogonal x-y coordinate system that has a certain magnitude |v|. Now lets say we obtain a new coordinate system by rotating the original coordinate system counter-clockwise around its origin. I know that with respect to the new coordinate system the vector would still have the same magnitude |v|. Thus, the vector would qualify as a rank1 tensor. This is intuitive and easy to understand. But, I often read about tensors that are applied with respect to different inertial reference systems. In this case, however, a velocity vector usually is not invariant with respect to two different inertial reference frames. But an acceleration vector is invariant and thus would qualify as a rank 1 tensor. So, where I am confused has to do with the term "change of coordinates". Is tensor invariance talking about invariance with respect to a change of coordinates as in the first example (a rotated coordinate system) or with respect to the second example (different inertial reference systems.) If someone could clarify this I would appreciate it.