dimension theorem
The dimension theorem is the result that makes 'dimension' a legitimate word: any two bases of the same vector space have the same number of elements. Without this you could not honestly say R^3 'is three-dimensional' — maybe some clever basis has four vectors? The theorem rules that out. Every basis of a given space has the same size, so that common size can be named once and for all: the dimension.
In finite dimensions the engine is the Steinitz exchange lemma: any linearly independent set can be no larger than any spanning set, because you can swap independent vectors into a spanning set one at a time without shrinking the span. Since a basis is both independent and spanning, two bases bound each other in size from both directions, forcing them equal. The proof is pure counting, no field-specific magic.
The same statement holds in infinite dimensions, with cardinality in place of a count: any two Hamel bases of V are in bijection — they have the same cardinality. The argument needs more care (and the axiom of choice), but the conclusion is identical: dimension, finite or infinite, is an intrinsic invariant of the space, blind to which basis you happened to pick.
Why this matters so much: dimension is the first and most robust invariant we have. Two spaces over the same field are isomorphic if and only if they have the same dimension — full stop. So dimension completely classifies vector spaces up to isomorphism, something no other branch of algebra enjoys so cleanly. Every later invariant (rank, multiplicities, signatures) is in some sense a refinement of this one foundational count.
The exchange lemma squeezes independent and spanning sizes together, pinning every basis to the same count.
The classification it yields is breathtakingly simple: over a fixed field, the ONLY invariant of a vector space is its dimension. Compare groups or rings, where two objects of the same size can be wildly different. Vector spaces are the rare algebraic structure with no hidden subtlety beyond a single number.