How is "what can I put in a matrix?" a fundamental question? Looking at the textbook, the author certainly covers the fundamental questions in vector spaces, linear maps, and decomposition/normal forms.
There's even a really great section on projective geometry that would have been helpful before seeing it quickly defined as homogenous coordinates before moving on to the definition as a quotient manifold of a sphere and Lie group.
Are you complaining about the axioms for a vector space? Because that list will exist for any LA class lol. That's like complaining that a college Calculus class is teaching the limit definition of the derivative.
I think (from only having skimmed the contents) that there is some handwaving around the definition of the characteristic polynomial. In order to properly define it, you need to allow matrices over arbitrary rings, not just fields.
There's even a really great section on projective geometry that would have been helpful before seeing it quickly defined as homogenous coordinates before moving on to the definition as a quotient manifold of a sphere and Lie group.
Are you complaining about the axioms for a vector space? Because that list will exist for any LA class lol. That's like complaining that a college Calculus class is teaching the limit definition of the derivative.