An abstract but still elementary application is that every field is a vector space over any of its subfields. In particular, every finite field $F$ is a vector space over its prime field, and so $|F| = p^n$ for some $n$ where $p$ is the characteristic of $F$. The same style of reasoning applied to finite extensions of $\mathbb{Q}$ gives negative solutions to the ancient problems of duplicating the cube and trisecting the angle with ruler and compass.

Galois theory has plenty of deeper applications of linear algebra to the study of field extensions. But the ones I mentioned are easily enough accessible that they could serve as motivation for the abstract approach to linear algebra.