isomorphic linear space

bernoli123

two linear spaces S and S1 over F are isomorphic if and only if there is a one-to-one
correspondence x↔ x1 between the elements x [tex]\in[/tex] S and x1 [tex]\in[/tex] S1
such that if x ↔ x1 and y ↔ y1 then x+y ↔ x1+y1 and ax ↔ ax1
(y [tex]\in[/tex] S , y1 [tex]\in[/tex] S1, a [tex]\in[/tex] F).
prove that two finite -dimensional spaces are isomorphic if and only if they are of the same dimension.
(The correspondence or mapping defining isomorphic linear spaces is called an
isomorphism).

Fredrik

Another way to say that, is that a vector space isomorphism is a linear bijection. So if U and V are vector spaces, and dim U=dim V=n (where n is some positive integer), you need to find a linear bijection T:U→V. I suggest that you use a basis for U and a basis for V to define a function T:U→V, and then show that T is injective, surjective, and linear.

If you want more help, you need to show us your attempt.