Any advantage when proving linear algebra statements without using bases? I always felt that proofs in linear algebra that do not assume the existence of bases seem more elegant but is there also something mathematically more valuable about these proofs?
I know that the existence of a basis requires the Axiom Of Choice in general, but I was more interested in the algebraic aspects.  For example, could it help to translate statements about vector spaces into stronger statements about modules?
 A: A linear algebra proof that avoids choosing a basis is primarily advantageous when one wants to use a vector space structure in a "canonical" or "coordinate-free" way.
From an algebraic perspective, if one can provide a basis-free proof of a property then it is likely to be a "canonical" property of the underlying structure, and knowing this can be more enlightening than ideas depending on the choice of a basis. For instance, one can prove quite easily that a finite-dimensional vector space and its dual are isomorphic by showing they are vector spaces of the same dimension (i.e. choosing bases). But this says nothing of substance when you consider the statement that all finite-dimensional vector spaces are isomorphic to $\mathbb{F}^n$ for some $n$.
On the other hand, one can prove that a vector space and its dual are isomorphic by using the action of the dual to construct an isomorphism without reference to a basis. This gives us a canonical way to pair a vector space and its dual, and this is more useful then just knowing that they are isomorphic due to having the same dimension. It actually sheds light on the relationship between the vector space and the dual, whereas a basis-dependent proof does not allude to this relationship at all.
