# Isomorphic linear space

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 $$\in$$ S and x1 $$\in$$ S1
such that if x ↔ x1 and y ↔ y1 then x+y ↔ x1+y1 and ax ↔ ax1
(y $$\in$$ S , y1 $$\in$$ S1, a $$\in$$ 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).