As linear transformations of space into itself, a very frequent operation, are described by a square matrix those matrices do show up more frequently. But map X into Y of different dimensions and you get a non-square matrix
Axler, and a few others, say this straight away in the intro. This makes things much simpler to understand instead of developing linear algebra historically from systems of linear equations.
there's a fantastic historiography of linear algebra and why we're stuck in this situation here: https://www.youtube.com/watch?v=C2RO34b_oPM
Calculus is also a bit confusing as for historical reasons we use a lot of notation from Newton-Leibniz informal infinitesimal approach, but we try to teach Bolzano-Weirstrass formal analysis, ending up with a confusing potpurri.
This is true, but it's also something that is arguably impossible on the one hand and "unwise" on the other.
You can easily map a higher-dimensional space into a lower-dimensional space, but you will irretrievably lose a lot of information when you do so.
And in the other direction, you can't actually map a lower-dimensional space into a higher-dimensional space with a matrix. The image of X can never have more dimensions than X does -- the choice to represent it with Y > X dimensions is just that, a representational choice. This idea is only meaningful in terms of the semantics behind the representation.
For example, an x-ray computed tomography (CT) image volume in 3D may be projected into various 2D synthetic planar x-ray projections (digitally reconstructed radiographs).
There are countless situations where projections are very useful.
The third dimension is discretized while the other two are continuous; the reconstruction consists of smoothing out the third dimension.
If I understand "unwise" to refer to a linear map into a lower-dimensional space: of course that's something you'll often want to do! Suppose, for example, that most of the interesting structure of your data is close to lying on a n-dimensional subspace of R^m, with n<m. Constructing a clever linear map from R^m to R^n can be very wise and useful!
> And in the other direction, you can't actually map a lower-dimensional space into a higher-dimensional space with a matrix. The image of X can never have more dimensions than X does -- the choice to represent it with Y > X dimensions is just that, a representational choice.
Any matrix involves a choice (of basis), so that complaint is moot. For sure you can have a linear map whose codomain is higher-dimensional. You are correct that the image can't be higher-dimensional, but that doesn't prevent the existence of such a map.