How can we describe isomorphism in a tangible way?

Question

What is the best way of describing isomorphism between two vector spaces? Is there a real life analogy of isomorphism?

Answer 1

Isomorphism is one type of homomorphism, so it is a structure-preserving map. Isomorphism is also bijective, so it's one-to-one correspondence. Put together, isomorphism a structure-preserving bijection. So if two spaces are isomorphic, they are mathematical equivalent.

Answer 2

An isomorphism, in almost every mathematical sense of the word, including between vector spaces, is a way of showing that two objects are some a fundamental way "the same".

An isomorphism between vector spaces, say $\phi : V_1 \to V_2$ will map any basis of $V_1$ to a basis of $V_2$ and the inverse will map a basis of $V_2$ to one in $V_1$. This is often a good way to view the isomorphism.

On the other hand, the image of any vector in $V_1$ will be a vector in $V_2$ which will interact with other vectors under addition and multiplication in $V_2$ in exactly the same way as the original would in $V_1$. So the isomorphism can also be viewed as a way of "renaming" $V_1$ as $V_2$ and preserving all important properties. This relabelling view may be the sort of real-world analogy you're looking for.