I just taught a module on Tensors in one of my physics courses. The mistake lots of people make is not show examples of matrices that are not tensors. But this is really difficult to do in physics courses because all physics matrix-like-objects must be tensors. Any theory that has non-tensor like objects in it will necessarily fail as soon as you change your coordinate system.

Thankfully, there is a great historical example of this. The electric field vector \vec{E} = (E_x, E_y, E_z), is not a tensor. It doesn't obey the tensor-transformation law. Similarly, the magnetic field vector is not a tensor. These are matrices, but not tensors.

As you know the Electromagnetic tensor [1] is the tensor that correctly transforms under coordinate transformation, and hence allows different observers to agree with each other.

[1] https://en.wikipedia.org/wiki/Electromagnetic_tensor

thanks for the examples. i vaguely recognize the term 'electromagnetic tensor', but i have to admit i didn't know that it was special in that way. i can barely spell 'tensor' at this point.

ps - one thing that always annoyed me was the limitation of linearity in so many of these models (which i totally understand why, but still). all the interesting real-world stuff happens non-linearly...

Indeed it is annoying. But the models are linear not because physicists mistakenly believe that the world is linear, but because in most cases linear models are the only ones that one can solve to get qualitative predictions out. Non-linear models can be constructed as well, and then numerically solved on computers to get exact answers. But a physicist is one who understand the essential qualitative features of the world, rather than one who can compute understanding-free numerical answers.

In the words of Dirac, "I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it." This usually only works if your equations are linear.

yah, as an engineer[0], i totally get the solvability angle, and even the physicist's core desire to be able to test (and predict via) the math rather than the physical manifestations (which may be impossible to test directly), but i'm eager to see us advance deeper into the non-linear, since that's where it gets really interesting. like, how do proteins really work? or multi-body energy fields? we're in the infancy of really understanding all this stuff. the future is stochastic and non-linear. in a thousand years, people might look back with amusement on how ignorant we were with our puny little linear models and deterministic computers. =)

[0]: but at this point, not really. even in grad school, i only did linear modeling, and relatively rudimentary ones, at that.