Linear algebra without division

So far we have been studying vector spaces. These are defined over a field, which in particular has division. But we might want to do look at a more general situation. For instance, we might want to think of $\mathbb{Z}^n$ as a “vector space” of “dimension” $n$ over $\mathbb{Z}$, even though $\mathbb{Z}$ is not a field. This can be made precise, although in this setting the trade-off is losing many the nice theorems we had in the case of fields.

Other than being just a generalization of the theory we have developed so far, this theory will also have an important application for vector spaces over fields. From a structure theory of modules over $k[t]$, we will immediately deduce Jordan normal form, in Section 4.4.