The deeper meaning of matrix transpose

Video Description

100k Q&A: https://forms.gle/dHnWwszzfHUqFKny7 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/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 the vector-covector correspondence: https://en.wikipedia.org/wiki/Raising_and_lowering_indices https://en.wikipedia.org/wiki/Minkowski_space#Raising_and_lowering_of_indices **ADJOINT** (a) Inner product (the prerequisite of even defining adjoints, the analog of dot products in Euclidean space): https://en.wikipedia.org/wiki/Inner_product_space (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/Hermitian_adjoint (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_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 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?) Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe! Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] For my contact email, check my About page on a PC. See you next time!