Learn Tensor Calculus: Understand Tensor Invariance

[e2m2a]

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 let's 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.

 

[fresh_42]

It is the same thing as with linear transformations, or vectors. If you change the basis, the representation changes, too, but not the vector or linear transformation. This is obvious in case of real objects and e.g. a rotation of it. If we change the measurement, then we get different numbers although neither object nor rotation has been changed. Now the difficulty is, that we describe object as well as rotation by numbers, and those do change. Thus the invariance is counterintuitive.

Have a look at https://www.physicsforums.com/insights/what-is-a-tensor/
with an example: Strassen's algorithm for matrix multiplication.