The unreasonable effectiveness of linear algebra.

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[MichaelPennMath]

To apply for an open position with MatX, visit www.matx.com/jobs.

[felipelopes]

For people who want to know more, what Michael Penn is hinting at is called Representation Theory. One very popular line of attack to classify mathematical structures is to represent them as compositions of linear transformations in vector spaces. In many cases of interest, you can prove that if you cannot find a representation with certain properties, then it means that the thing you are trying to study does not have an important property. And since studying representations is much easier than studying the abstract structure, it simplifies things a lot.
That's how Fermat's Last Theorem was ultimately conquered. They reduced the problem to the nonexistence of a given structure, and through some long arguments could reduce it to properties of the representations, which could be brute forced to prove no solution would exist.

[godelianconfucianism]

"If you can reduce a mathematical problem to a problem in linear algebra, you can most likely solve it, provided you know enough linear algebra". This was a quote in the preface of Linear Algebra and its Applications by the great mathematician Peter D. Lax. It was my first book on the subject and that sentence stuck with me ever since

[michakuczynski]

My quantum mechanics professor once mentioned that "we are very lucky that the fundamental laws of nature are expressed using the language of linear algebra". This video really changed my perspective on this matter...

[boomerzilean]

I think when you learn linear algebra as a student you don't really get this impression, but I think it's an important thing to realise when you study mathematics, that linear algebra is just conceptually extremely easy and is basically "solved" as a subject

[aweebthatlovesmath]

Algebra is a really important subject of math and almost everything in algebra can be understood with linear algebra via representation theory this makes linear algebra a really powerful tool!

[janvesely]

The best application of linear algebra is certainly the functional analysis. It transfroms the mess of differential/integral equations into something really elegant and easy to be used.

[ketv]

I've done a lot of numerical analysis in a long career. I've long claimed that 90% of the job is finding the integral transform that maps your impossible problem into linear algebra, and then letting a computer do the linear algebra. If asked what piece of the subroutine libraries that I would re-implement first if I didn't have them, I'd have to say that it's the singular value decomposition. It's the Swiss Army Knife of numerical linear algebra.

[gslpkoc]

Wow! Great overview! My favorite applications of linear algebra: spherical geometry (makes the equations intuitive), Fourier analysis, multivariate Gaussian distributions, affine transformations of random variables, linear regression, and engineering problems combining some of the above, especially when the matrices can be manipulated to make the solution methods (almost) unreasonably elegant! 🙂

[mathisnotforthefaintofheart]

I found the idea of using Cayley Hamilton to find the four square roots of a 2 by 2 matrix stunning. Because the algebra behind it ultimately is so easy...Linear is a must for every math inclined person.

[ejuicy]

I like to tell people who don't know so much math that pure maths can be divided into two topics:

1. Linear Algebra
2. Turning problems into Linear Algebra

[PhilBoswell]

At 16:50 should the adjacent matrix be adjusted a bit? It seems to suggest that node 1 is only connected to itself and node 5, missing out the connection to node 2, and that node 2 is connected to itself…

[scottcentoni]

My favorite application of linear algebra is quantum mechanics. Quantum chemistry basically is a huge eigenvalue problem. If you use a plane wave basis with periodic boundary conditions, you can do some of the calculations much more efficiently in momentum space using a fast Fourier transform.

[JalebJay]

I remember using it for mono directional paths. With the trace of A^n being the number of ways to complete a cycle after n steps. Useful in one of my classes back in 2013.

[muhammadkumaylabbas]

Very fascinating. As usual a super interesting video!

[TheLuckySpades]

In a bunch of classes we would reduce parts of the problems to Linear Algebra and the proof would then be written as "proof via LinAlg"

[kylebowles]

Finally a video i could keep up with! Theres a small error in the adjacency matrix on column 2 but this was a great video. I recently used linear algebra to least squares fit a function on a non euclidian manifold. Linear algebra is unreasonably effective even in intrinsically nonlinear spaces lol

[jarnorajala]

Besides the adjacency matrix, a graph can be represented using the closely related Laplacian matrix. This has some mind-blowing applications. Like if you take the two eigenvectors corresponding to the two largest eigenvalues (by absolute value) of the Laplacian and use them as arrays of X and Y coordinates for the nodes, you get a really nice 2D representation of the graph that happens to be the solution to a particular optimization problem where the edges are springs.

[icenarsin]

Wow!!! I never saw this connection before. Integrating, by using and inverse matrix!!! So awesome. Thank you!

[tommychau]

Days ago, I was thinking of the volume of a slanted/oblique cone. If it was a cylinder, it would be obviously the same as a regular cylinder by thinking a stack of disks pushed sideways.

After little digging, it’s called Cavalieri’s Principle and It should work for cones as well. So from other perspective, I tried to write this pushing stack of disks into matrix, which formed something called “shear matrix”. Amazingly, the determinant is 1, meaning there is no impact on the volume!