Looking back on my math education, I noticed that even though the trace of an endomorphism came up a lot, I'd be hard pressed to give a good description of what the trace really means. I'm looking for some good intuition about its meaning.
To elaborate on what I'm looking for: if I forgot the rigorous definition of the determinant, I could rebuild it from scratch without reference to a basis because I know that it's supposed to measure the change in volume and orientation of a parallelepiped under a linear transformation. For the same reason, I can quickly tell that it is independent of basis, multiplicative, and determines wether the endomorphism is injective or not, all without doing any calculations. I want something similar for the trace. It doesn't need to be geometric, but I want to know what the trace tells us about how the endomorphism acts.