Question

Let V be a vector space with ordered basis B = (b1, . . . , bn). Does the basis having n elements imply that V is the coordinate space R^n?

Answer 1

No, Having n elements in basis does not imply that V is the coordinate space R^n. For example consider Space of polynomial of degree less than or equal to n-1 over the field of real numbers. then the space is n-dimensional space, and basis have element {1,x,x^2,...,x^n-1} which are n in numbers but space is not R^n.

Although there is a celebrated theorem (I am not mentioning it in the exact form and dropping few hypotheses) and finite-dimensional vector space of dimension n is isomorphic to R^n

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