How are vector spaces viewed as universal algebras?

Proof in vector spaces, please help?

• Let V be a vector space. Prove that if it is possible to find m vectors in V which are linearly independent, and n vectors which span V, then m must be less than, or equal to, n. Any help would be appreciated.

• Answer:

  If V is infinite dimensional, then no finite number n of vectors can span V and the result is trivial (m < ∞). Otherwise, let d = dim(V). Every spanning set must contain at least d vectors (by the definition of dimension.) So if n vectors span V, we have n ≥ d. As every linearly independent set can have at most d vectors (again by the definition of dimension), if m vectors are linearly independent, then m ≤ d. So m ≤ d ≤ n giving the required conclusion m ≤ n.