What Defines Isomorphism in Different Mathematical Structures?

[sihaia]

Homework Statement

(i) Set of all row vectors: (a1,...,an), aj in K; addition, multiplication defined componentwise. This space is denoted as Kn.
(ii) Set of all real valued functions f(x) defined on the real line, K = R.
(iii) Set of all functions with values in K, defined on an arbitrary set S.
(iv) Set of all polynomials of degree less than n with coefficients in K.

Homework Equations

1) Show that (i) and (iv) are isomorphic
2) Show that if S has n elements, (i) is the same as (iii)
3) Show that when K = R, (iv) is isomorphic with (iii) when S consists of n distinct points of R.

The Attempt at a Solution

I've solved 1), but I cannot solve others. I think that problem is that I don't understand definition of (iii).

Could someone please help me?

 

[Office_Shredder]

An example of (iii). If S is the rational numbers, then (iii) would be the set of functions from the rational numbers to the real numbers. Examples would be f(x)=x, f(x)=x², f(x)=sin(x), where x is a rational number

If S is the set containing just the numbers 1,4,7 and 9, then f(x) only takes four values. Because you only have f(1), f(4), f(7) and f(9). So if K is the real numbers again, a sample element of S would be the function f(x) with f(1)=2, f(4)=pi, f(7)=0 and f(9)=pi