We consider the field of "usual" linear algebra.
Q. Which aspects of it can be carried out without the Axiom of Choice?
Q. Do interesting "exotic" phenomena appear in presence of (some instance of) the negation of the Axiom of Choice?
Without Choice, vector spaces may have a basis (hence, in particular, be dimensional) or not and hence be adimensional. [As Andreas Blass observes in the comments, the terminology "dimensional/adimensional" should rather be used to denote the property of having all bases of the same cardinality, rather than just having a basis, as there are vector spaces with two bases of different cardinality]
Q. Could the following property of a vector space $V$
Property ($\star$) Every injective endomorphism of $V$ is an automorphism.
be a valid substitute for finite-dimensionality for the class of not-necessarily-dimensional vector spaces over a field? Would the linear algebra of vector spaces verifying ($\star$) be reasonably similar to the usual one for finite dimensional spaces?