free vector space on a set

Given any set S — of letters, animals, points, anything at all — the free vector space F(S) over a field F is the space of formal finite linear combinations of the elements of S. A typical element looks like 3*cat - 2*dog + 7*fish: the symbols cat, dog, fish are treated as independent basis vectors, and the coefficients come from F. You are manufacturing a vector space whose basis is, by construction, exactly the set S you started with.

Concretely, F(S) is the space of functions S -> F that are zero except at finitely many points; the element s of S corresponds to the function that is 1 at s and 0 elsewhere. Addition and scaling are pointwise. The set S sits inside F(S) as a basis, so dim F(S) is the cardinality of S — you have turned a bare set into a vector space in the most economical way, adding no relations among the elements beyond what the axioms force.

The deep characterization is the universal property, and it is what 'free' really means. Any function from S into any vector space W extends to a UNIQUE linear map F(S) -> W. In words: to define a linear map out of the free space, you need only say where the basis elements (the original symbols) go, with total freedom — no constraints to honor. The free space is the most efficient vector space generated by S, imposing zero relations beyond linearity.

Why bother building a space this way? Because it lets you linearize any set. Formal sums of paths build chain groups in topology; formal sums of basis states build quantum state spaces; formal sums of group elements build the group algebra. Whenever you want to 'take linear combinations' of objects that are not yet vectors, the free vector space is the canonical machine that grants them that power — and the universal property guarantees you have added nothing extraneous.