Sets in Linear Space(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to show the set of all row vectors in some set K with dimension n is the same as the set of all functions with values in K, defined on an arbitrary set S with dimension n. I am using isomorphism to show this, but I can't determine how to show that the isomorphism is onto.

My work so far:

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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# Homework Help: Linear Algebra Isomorphism

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