/sci/ - Science & Math

How do I do exercise 4 -- show that if S has n elements, (i) and (iii) are isomorphic.
Let K be either the field R of real numbers or the field C of complex numbers.

(i) set X of all row vectors: [math] (a_1,...,a_n), a_j \in K [\math]
(iii) set Y of all functions with values in K, defined on an arbitrary set S.

My interpretation of isomorphic: there exists some function, [math]\phi:X \rightarrow Y[/math] where X and Y are two linear spaces over the field K, and k an element of K, such that
(a) [math]\phi(X_1 + X_2) = \phi(X_1) + \phi(X_2)[\math] and
(b) [math]\phi(kX) = k\phi(X)[\math]

Here, would phi would be the function to take all the a_j's from X and map them into functions of a_j in Y. e.g., [math] \phi((a_1,...,a_n)) = \{f(a_j)\} [\math] ? If this was so, I don't think it'd be isomorphic.