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?

4more comments