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

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Isomorphic vector spaces (or groups or whatevre) are "the same for all that matters". –  Hagen von Eitzen Oct 28 '12 at 14:34
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I say tomato you say tomayto... –  fretty Oct 28 '12 at 14:36
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Both are labels for the same real world object. An isomorphism of two structures is basically a relabelling in which operations are preserved too. –  fretty Oct 28 '12 at 14:37
    
Natural isomorphism and uniqueness up to unique isomorphism (Mazur's "When is one thing equal to another thing?") should also be kept in mind. –  alancalvitti Oct 28 '12 at 14:55

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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.

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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.

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