Spacetime and Linear Algebra

It is well known that Newtonian mechanics and electrodynamics are fundamental theories of physics and also theories of our physical spacetime. Any theory of spacetime is, of course, a geometrical theory. As we already saw, linear algebra is, in many aspects, also geometric. The surprise is that linear algebra, along with some groupGroup theory, allows us to describe spacetime geometry in Newtonian mechanics and electrodynamics. Consequently, to describe this surprise, we will show the relation of linear algebra with the two central principles of Newtonian mechanics and electrodynamics, in particular with the law of inertia and the relativity principle, with linear algebra. For thousands of years, it was thought to be an absolute convention that space and time are a priori given and physics itself had to be formulated in this context. This, by itself a very plausible position, was also taken in science for more than two thousand years. It has its roots in the fascination and power of the Euclidean axioms. But since Gauss and Riemann we had suspected, and since Einstein we have known that physics is the one that determines the structure of spacetime. This is demonstrated in this chapter.