Generalize Example 30.2 to obtain the vector space Fⁿ of ordered n-tuples of elements of F over the field F, for any field F. What is a basis for Fⁿ?

Data from in Example 30.2

Consider the abelian group (R_n, +) = R x R x ··· x R. for n factors, which consists of ordered n-tuples under addition by components. Define scalar multiplication for scalars in R by rα = (ra₁, .... ra_n) for r ∈ R and α = (a₁, ··· , a_n) ∈ Rⁿ. With these operations, Rⁿ becomes a vector space over R The axioms for a vector space are readily checked. In particular, R² = R x R as a vector space over R can be viewed as all "vectors whose starting points are the origin of the Euclidean plane" in the sense often studied in calculus courses.