Transposing isn't just about swapping rows and columns, it's more about changing perspectives to get the same measurements. By understanding the general idea of transposing a linear map, we can use it to visualize transposition much more directly. We will also rely heavily on the concept of covectors, and touch lightly on metric tensors in special/general relativity and adjoints in quantum mechanics.
To my knowledge, this way of visualizing the transposition is original. Most people use SVD (singular value decomposition) for such visualization, but I think it's much less direct than this, and SVD is also mainly used for numerical methods, so it seems somewhat unnatural to use a numerical method to explain linear transformations (of course, SVD is extremely useful). Please let me know if you know of other people having this specific visualization.
The concept I'm introducing here is usually called "pullback" (and in fact the original linear transformation would be called "pushforward"), but as shown in the video, another way of thinking about transposition is the notion of "adjoint" .
Remarks:
(1) I call covectors a "measuring device", not only because the representation of covector levels looks like a ruler when you take a strip of the plane, but also because of its connections to quantum mechanics. A “bra” in quantum mechanics is a covector and can be thought of as a “measurement”, in the sense of “how likely is it that you measure this state” (sort of).
(2) I deliberately do not use line vectors to describe covectors, not only because it only works in finite-dimensional spaces, but also because it is awkward for order when we say that a transposition matrix *acts* on the covector. We usually apply transformations to the *left*, but if you treat the covector as a row vector, you need to act on the transpose matrix to the *right*.
(3) You can do a sort of "exercise" to check this transpose visualization for all (non-singular) matrices, but I think the algebra is a bit too tedious. This is why I spent a lot of time talking about the overview of transposition – to make the explanation as natural as possible.
Further reading:
**GENERAL**
(a) Transposition of a linear map (Wikipedia)
https://en.wikipedia.org/wiki/Transpose_of_a_linear_map
(b) Vector space not isomorphic to its dual (for infinite-dimensional vector spaces):
https://math.stackexchange.com/questions/35779/what-can-be-said-about-the-dual-space-of-an-infinite-dimensional-real-vector-spa
https://math.stackexchange.com/questions/58548/why-are-vector-spaces-not-isomorphic-to-their-duals/58598#58598
**RELATIVITY**
(a) Metric/inverse metric as vector-covector correspondence: https://en.wikipedia.org/wiki/Raising_and_lowering_indices
https://en.wikipedia.org/wiki/Minkowski_space#Raising_and_lowering_of_indices
**DEPUTY**
(a) Dot product (the very prerequisite for the definition of adjoints, the analogue of dot products in Euclidean space): https://en.wikipedia.org/wiki/Inner_product_space
(b) Adjoints (another way of thinking about transpositions, but I think it's mainly the complex analogue of transposition): https://en.wikipedia.org/wiki/Hermitian_adjoint
(c) Reisz's representation theorem (more relevant for the adjoints, but regarding 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_representation_theorem
(d) Self-adjoint operators (Hermitian operators in QM, but also useful in Sturm-Liouville theory in ODEs):
https://en.wikipedia.org/wiki/Self-adjoint_operator
Video chapters:
00:00 Presentation
00:56 Chapter 1: Overview
04:29 Chapter 2: Visualizing covectors
09:32 Chapter 3: Visualizing the transposition
16:18 Two more examples of transposition
19:51 Chapter 4: Subtleties (special relativity?)
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