If we restrict ourselves to the field of linear algebra,
my personal point of view, which I do not want to force on anybody,
is that one should never use bases, matrices, or coordinates.
The main reason is that you lose geometric intuition whenever you introduce a basis,
and geometric intuition in linear algebra is extremely important for me,
not only in definitions but also in theorems and their proofs.
When I learned linear algebra I made sure I understood the geometric meaning of every
single definition, theorem, and proof.
For example, an element of a vector space is a vector or a 1-dimensional subspace with an oriented metric,
an element of the dual vector space is a hyperplane with an oriented metric on its “complement”, i.e., the factor space,
an element of the exterior algebra is a formal sum of vector subspaces
(dimension equals degree) equipped with oriented metrics,
an element of the exterior algebra of the dual space is a formal sum of vector subspaces (codimension equals degree)
with an oriented metric on their “complement”, i.e., the factor space,
the exterior product of two elements of the exterior algebra is the direct sum of the corresponding spaces (or zero if they have nontrivial intersection) with the obvious choice of an oriented metric,
the inner product of an element of the exterior algebra and an element of the dual exterior algebra
is the intersection or the sum (depends on the type of the inner product) of the corresponding subspaces
with the obvious choice of an oriented metric, Hodge star is a particular case of the previous construction
(if you have a subspace with an oriented metric and also an oriented metric on the entire space then you
can canonically produce an oriented metric on the “complement”, i.e., the factor space),
trace and determinant also have an obvious geometric meaning in this framework etc. etc. etc.
All of this is fully rigorous and all theorems and their proofs become trivial once you have a geometric intuition
for all definitions, and you don't need any bases, coordinates, or matrices, even when you prove something.
Ironically, the best source for geometric intuition in linear algebra for me was Bourbaki's Algebra,
which is often blamed for its abstractness.
Actually it is the only source known to me that (indirectly) explains the geometric meaning of exterior algebra
(please tell me if you know other sources).
I badly want to see a sufficiently advanced textbook on linear algebra that at least includes all the notions mentioned above (and many others, of course)
and satisfies the following two conditions:
(1) It explains the geometric meaning of every single definition, theorem, and proof (or states them in such
a way that their geometric meaning is evident);
(2) It never uses bases, coordinates, or matrices and does not even define these notions.