Redundant axioms in teaching linear algebra

What is an axiom? The meaning of the term ‘axiom’ varies between different fields of study. In philosophy it means a statement that is accepted without argument, while in physics the term ‘postulate’ is used and it means a theory that was verified in an experiment, and will be considered as true unless it is disproved by other experiments. In mathematics the notion of ‘axiom’ is used in two related but distinguishable meanings: ‘logical axioms’ and ‘non-logical axioms’. Logical axioms are certain statements that are always true, and from them all tautologies of the language can be derived. Non-logical axioms are statements that are taken to be true, to serve as a premise or starting point for further reasoning and arguments.