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Every Finite-Dimensional Space Is Just R^n in Disguise

The payoff: choose a [[hamel-basis|basis]] and a finite-dimensional space becomes a coordinate space. The [[dimension-theorem|dimension theorem]] says one number is the *complete* invariant — any two spaces of the same dimension are [[isomorphism-of-spaces|isomorphic]] via a [[coordinate-isomorphism|coordinate isomorphism]].

A basis turns any space into coordinates

Pick a basis B = (b_1, ..., b_n) of a finite-dimensional space V (a finite [[hamel-basis|Hamel basis]]). Every v is a unique linear combination v = c_1 b_1 + ... + c_n b_n, and the coordinate vector [v]_B = (c_1, ..., c_n) lives in R^n. The map v -> [v]_B is the [[coordinate-isomorphism|coordinate isomorphism]]: it preserves addition and scaling perfectly.

P_2 (degree <= 2 polynomials),  basis B = (1, x, x^2):

   3 - 5x + 2x^2   ->   [v]_B = (3, -5, 2)  in R^3
   1 + x^2         ->          (1,  0, 1)

   add the polynomials  <=>  add the coordinate triples
   (3-5x+2x^2)+(1+x^2) = 4 -5x +3x^2  ~  (3,-5,2)+(1,0,1)=(4,-5,3)

P_2 'is' R^3 once you fix the basis -- different costume, same algebra.
Choosing a basis is choosing a coordinate isomorphism V -> R^n.

Isomorphism: same structure, different labels

Two spaces V and W are [[isomorphism-of-spaces|isomorphic]] (V ≅ W) if there is a bijective linear transformation T: V -> W. Isomorphic spaces are the *same* as vector spaces — every linear-algebra fact about one holds in the other, just with renamed elements. An isomorphism is a perfect dictionary between two structures.

  1. Pick a basis of V (size n) and a basis of W (size m).
  2. Each gives a coordinate isomorphism: V ≅ R^n and W ≅ R^m.
  3. R^n ≅ R^m holds iff n = m. So V ≅ W exactly when dim V = dim W.

The dimension theorem: one number says it all

Here is the field's big payoff — the [[dimension-theorem|dimension theorem]]: for finite-dimensional spaces over a fixed field, dimension is a complete invariant. Two such spaces are isomorphic if and only if they have the same dimension. One natural number classifies *every* finite-dimensional vector space, completely and with nothing left over.

Two caveats sharpen the result. First, the field matters: C is a 1-dimensional space over itself but a 2-dimensional space over R — a field extension viewed as a vector space. Second, the theorem stops at 'finite-dimensional'. An infinite-dimensional space like the function space F(R), or the free vector space on an infinite set, still has a Hamel basis (assuming choice), but no single integer pins it down. Within the finite world, though, you now know the whole story: think structurally, and every space is just a coordinate space up to isomorphism.