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TLDR: the author is asking an honest question, not trying to say that non square matrices have little use in linear algebra. As linear transformations of space
by ptero 6y ago
TLDR: the author is asking an honest question, not trying to say that non square matrices have little use in linear algebra.
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
- nextos 6y agoI think the key is that linear algebra textbooks do not tell students straight away that matrices are encodings of linear maps. Hence a square matrix is an encoding of a map that preserves dimensions. 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.
- dnautics 6y agoI happened to be the other way, I learned first that matrices were linear transformations and, being thick-headed, it took me a really long time to understand that matrices could also be used as "dumb excel spreadsheets". 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 https://www.youtube.com/watch?v=C2RO34b_oPM
- nextos 6y agoA worse situation is that of statistics, where many textbooks insist on using approximations to the normal which obscure concepts and are not needed in the age of computer-based inference [1]. 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. [1] https://escholarship.org/uc/item/6hb3k0nz https://escholarship.org/uc/item/6hb3k0nz
- dnautics 6y agoI didn't mind the issue in calculus. The first thing we did in analysis after being introduced the Bolzano weierstrass form was to prove that it was equivalent to epsilon delta proofs.
- thaumasiotes 6y ago> But map X into Y of different dimensions and you get a non-square matrix 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.
- afwaller 6y agoSometimes losing that information is very useful however, to arrive at a representation that has a specific meaning. 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.
- thaumasiotes 6y agoI don't disagree, but I think my comment helps to illustrate a reason why square matrices really are more special than other matrices.
- unsrsly 6y agoIn fact, the 3D CT image itself is reconstructed from projections! See https://en.wikipedia.org/wiki/Tomographic_reconstruction https://en.wikipedia.org/wiki/Tomographic_reconstruction
- thaumasiotes 6y agoYes, but this is a case where you have three dimensions in the input and also three dimensions in the output. (You're assembling the 3D image from different 2D slices with different depth coordinates.) The third dimension is discretized while the other two are continuous; the reconstruction consists of smoothing out the third dimension.