I come from the applied math and statistics world, but I was talking to my friend who comes from the pure math and number theory world—in particular Galois representation theory.
I mentioned something about the confusing definition of "matrices" in textbooks. Many textbooks talk about a matrix as the solution to a linear system of equations, or other abstract descriptions. The definition that I have always found useful, was the sense that matrices rotate and scale vectors through linear transformations.
Now, coming from the pure math side, my friend said that this definition was not accurate. I am trying to paraphrase some of his comments, but he said that in higher dimensions, matrices can stretch and rotate vectors only locally. He also said that it depends on what vectors the matrix acts upon.
His response threw me for a loop and I was trying to understand how to resolve his statements. First, is my understanding of matrices incorrect?
It is fine if this idea of rotating and stretching a vector is incorrect, but I am not sure what the alternative definition of a matrix would be. Like when my friend says that a matrix may only stretch/rotate a vector locally, I am trying to think of a practical example of this kind of phenomena.
I will follow up with my friend on what he means, but I was hoping someone could set me straight on how I should think of a matrix, or the definition of a matrix.