Benefits of Vector Axioms
Vector spaces are important as a collection of properties that generalize many mathematical ideas.
They abstract some common structures from these ideas and unify reasoning.
It is the axiomatic method at its best.
Often mathematical problems arise where no obvious geometry or n-tuple of reals is in view and then it is harder to notice the vector space—and it is in these situations that vector theorems may provide the biggest boost.
There is a vast literature on matrices, which are merely transformations on vector spaces with identified basis elements.
Vector spaces viewed axiomatically can often harness these theorems if they are expressed in coordinate free notions and often the intuition is liberated from details of coordinates.
Often it is necessary to move back and forth between matrix discipline and the abstract space to see the whole picture, and especially to harness the computer in numerical applications.