Is there any application for a linear map from a lower dimensional space to a higher dimensional space? Does it make any sense to consider a linear map from a lower dimensional space to a higher dimensional space, i.e. matrix representations of linear maps $A$, where $A \in \mathbb{K}^{m \times n}, m>n$. Intuitively, I see that $A$ can only map to a linear subspace of $\mathbb{K}^m$. Do these maps ever pop up?
I guess it is pointless to consider this map right? We could simply align the basis in $\mathbb{K}^m$ to span the range of $A$ and then consider $A$ as a map from $\mathbb{K}^n$ to $\mathbb{K}^n$. Is this the right line of thinking?
Thanks.
 A: It is a perfectly sensible thing to talk about, and there are plenty of uses for such maps. Take a look at overdetermined systems.
For another picture, think of how many different ways you can find linear maps of the plane $\Bbb R^2$ into $\Bbb R^3$. There are lots of different ways to "position" the plane in the volume.
A: It's not pointless. There are some pretty reasonable ways to setup a non-trivial use for this. 
Consider that we might 'perturb' $A$ by a little bit. That is, consider $A(\lambda): U \to \mathbb{K}^{m\times n}$ where $U$ is maybe a subset of complex numbers. Sure for any particular value of $\lambda$ we can pick an $n$ dimensional subspace, but as $\lambda$ changes so does that space. We might want to study how badly that space varies for small changes in $\lambda$. 
So while your point is fair for any particular such matrix (as far as geometry goes), generalizing a little can create more interesting behavior.
A: Here comes a list:


*

*Why not? It is a straightforward "generalization" with no problem at all.

*Surely it seems reasonable you to consider a smooth map from a manifold (read "space" if you are not familiar with the term) of lower dimension to one of higher dimension. For example, a curve is a map $\mathbb{R} \rightarrow X$. Its derivative is a linear map from a low dimension to a higher dimension.

*The inclusion of a subspace in another is a linear map, and a pretty natural one to consider.

*As said in another answer, overdetermined systems commonly appear in applications.

*etc

