For an unconventional but beautiful presentation of the subject, I highly recommend "Wild Linear Algebra" by the brilliant Norman Wildberger:

https://www.youtube.com/watch?v=yAb12PWrhV0&list=PLBQcPIGljH...

It starts with the axioms of being able to draw one line parallel to another, and a line through point, and builds up everything from there. No labeled Cartesian axes. Just primitive Euclidean objects in an affine space.