The deeper meaning of matrix transpose

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Transpose isn’t just swapping rows and columns - it’s more about changing perspective to get the same measurements. By understanding the general idea of transpose of a linear map, we can use it to visualise transpose much more directly. We will also heavily rely on the concept of covectors, and touch lightly on metric tensors in special/general relativity, and adjoints in quantum mechanics.

As far as I know, this way of visualisation of transpose is original. Most people use SVD (singular value decomposition) for such visualisation, but I think it is much less direct than this one, and also SVD is mostly used for numerical methods, so it feels somewhat unnatural to use a numerical method to explain linear transformations (though, of course, SVD is extremely useful). Please let me know if you know that other people have this specific visualisation.

The concept I am introducing here is usually called a “pullback” (and actually the original linear transformation would be called “pushforward”), but as said in the video, another way of thinking about transpose is the notion of “adjoint”.

Notes:
(1) I am calling covectors a “measuring device”, not only because the level set representation of covectors looks like a ruler when you take a strip of the plane, but also because of its connections with quantum mechanics. A “bra” in quantum mechanics is a covector, and can be thought of as a “measurement”, in the sense of “how likely will you measure that state” (sort of).

(2) I deliberately don’t use row vectors to describe covectors, not only because this only works in finite-dimensional spaces, but also because it is awkward for the ordering when we say a transpose matrix acts on the covector. We usually apply transformations on the *left*, but if you treat the covector as a row vector, you have to act the transpose matrix on the *right*.

(3) You can do the sort of “exercise” to verify this visualisation of transpose for all (non-singular) matrices, but I think the algebra is slightly too tedious. This is the reason why I spent a lot of time talking about the big picture of transpose - to make the explanation as natural as possible.

Further reading:

*GENERAL*

(a) Transpose of a linear map (Wikipedia)
https://en.wikipedia.org/wiki/Transpo...

(b) Vector space not isomorphic to its dual (for infinite-dimensional vector spaces):
https://math.stackexchange.com/questi...
https://math.stackexchange.com/questi...

*RELATIVITY*

(a) Metric / inverse metric as the vector-covector correspondence: https://en.wikipedia.org/wiki/Raising...
https://en.wikipedia.org/wiki/Minkows...

*ADJOINT*

(a) Inner product (the prerequisite of even defining adjoints, the analog of dot products in Euclidean space): https://en.wikipedia.org/wiki/Inner_p...

(b) Adjoints (another way of thinking about transposes, but I think this is mostly about the complex analogue of transpose): https://en.wikipedia.org/wiki/Hermiti...

(c) Reisz representation theorem (more relevant to adjoints, but in regards to the statement that “we choose certain covectors to act on”: here, it is the “continuous” dual, very relevant in QM): https://en.wikipedia.org/wiki/Riesz_r...

(d) Self-adjoint operators (Hermitian operators in QM, but also useful in Sturm-Liouville theory in ODEs):
https://en.wikipedia.org/wiki/Self-ad...

Video chapters:
00:00 Introduction
00:56 Chapter 1: The big picture
04:29 Chapter 2: Visualizing covectors
09:32 Chapter 3: Visualizing transpose
16:18 Two other examples of transpose
19:51 Chapter 4: Subtleties (special relativity?)

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