Vector spaces are the central objects of linear algebra. We define them abstractly over arbitrary fields, study their structure through subspaces and bases, and discover that dimension is the fundamental invariant.
By working abstractly, theorems apply to ℝⁿ, function spaces, polynomial spaces, and more—all at once
Two finite-dimensional spaces over the same field are isomorphic iff they have the same dimension
A basis gives coordinates, but there are infinitely many bases—coordinate-free thinking is often cleaner
Direct sums combine spaces, quotients collapse subspaces—these constructions are everywhere in algebra