# Isomorphism An **[Isomorphism](https://en.wikipedia.org/wiki/Isomorphism)** is a *structure-preserving* mapping between two structures of the same type that can be reversed by an inverse mapping. ### Isomorphism vs Bijection If you are talking just about sets, with no structure, the two concepts are identical. Usually the **term "isomorphism" is used when there is some additional structure** on the set. For example, if the sets are vector spaces, then an isomorphism is a bijection that preserves vector addition and scalar multiplication. So an example of a bijection that is not an isomorphism would be: $f(x, y) \rightarrow (x + 1, y)$ Since it is clearly **one to one** and **onto** (and hence a **bijection**), but it does not preserve vector addition. --- Date: 20220613 Links to: [Mathematics MOC](Mathematics%20MOC.md) [Linear Algebra](Linear%20Algebra.md) Tags: #review References: * []()