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A first-time student in linear algebra definitely does not need to hear anything about finite fields, though. I don't think anyone outside of certain mathematics subfields needs to.

Geometric interpretations have a place in learning: until one is ready to understand another interpretation that is more complete, better, yet more complicated.

You don't start kids with complex numbers until they can handle reals. You don't even start negative numbers until they can handle positive numbers.

Given a vector space V, form its top wedge power. This is one-dimensional. And endomorphism of V induces an endomorphism of the top wedge power. But an endomorphism of a 1-dimensional vector space is a number. This number is the determinant.

This is probably the correct generalization of the volume definition.

> linear functions from 2^n to 3^n

Say what?

2^n is {0, 1}^n, the space of n-tuples with elements from the smallest field. The field is also known as GF(2), Z/2Z, or F₂.