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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).