Prove that the set of all functions f : S -> K defined on a set S of n elements

Question

Prove that the set of all functions f : S -> K defined on a set S of n elements

S = {a1,a2,...,an}

and valued in a field K form a vector space over K under the operations of addition of functions and multiplication by a scalar from K.

Explanation / Answer

We show Linearity: let c1, c2 be in K and let g,h be in G f(c1*g + c2*h) = (c1*g(s1) + c2*h(s1),...,c1*g(sn) + c2*h(sn)) = c1*(g(s1),...,g(sn)) + c2*(h(s1),...,h(sn)) = c1*f(g) + c2*f(h) Hence this proves the preservance of form for both addition and multiplication MORE DETAILED SCRIPT : ========================== G := Functions with values in K f: G -> K (g(s1),...,g(sn)) one to one f(G) = f(H) (g(s1),...,g(sn)) = (h(s1),...,h(sn)) therefore, g(s1) = h(s1), ..., g(sn) = h(sn) preserves structure f(c1*G + c2*H) = (c1*g(s1) + c2*h(s1),...,c1*g(sn) + c2*h(sn)) = c1*(g(s1),...,g(sn)) + c2*(h(s1),...,h(sn)) = c1*f(G) + c2*f(H)

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