I went though the slides. Super fun material! I've seen all the methods long ago, and much deeper than in the slides, and published on some of the most advanced material, and much more, but, still, it was fun material because of the many examples and really good graphs.

From their other books, clearly they are real experts. The slides, then, are a careful path where minimal theory gives a LOT of nice applications. The theory they give is nearly always so simple that they are able, in just a few lines, to give essentially the proofs, nearly always.

E.g., I never saw any mention of convexity, and these two guys are right at the top of experts on theory and applications of convexity, so that it is clear that they tried hard to get lots of applications from minimal theory.

They did next to nothing on numerical stability -- some mention might have been good.

There's a still easier derivation of the least squares normal equations based on perpendicular projections -- they might have included that. That is, if drop a golf ball to the floor, the line to the floor and the shortest distance to the floor is the line perpendicular to the floor. This fact generalizes.

They have illustrated a nice, general lesson: Can do such applications with just finite dimensions and/or discreteness. Can do more theory with continuous instead of discrete values and infinite instead of finite dimensions. But, then, even with the extra theory, often challenging, commonly for the computing are back to discreteness and finiteness. Sooooo, just omit the more advanced theory and just stay discrete and finite throughout -- that's one of the themes in the slides.

With this theme, the slides are able to do at least something interesting and potentially valuable from stacks of texts in pure and applied math, statistics, and more with just a few slides, simple math, nice graphs, and a few words. Nice.

E.g., they did a lot of applied statistics without mentioning probability theory! How'd they do that? They just stayed with the data and omitted describing the probabilistic context from which the data was samples or estimates. Cute. But, readers, be warned -- the probabilistic context should not be neglected; eventually should learn that, too.

Another cute omission of theory -- vector subspaces and the, really, the axioms of a vector space. E.g., that "floor" I mentioned above is such a subspace. How'd they do that? They just stayed with the basic example vector spaces they had in mind and managed to avoid talking about subspaces.

At one point they touched on determinants for the 2 x 2 case, mentioned that that result is important (should be remembered or some such), that there is a more general approach that don't have to remember!!! Determinants have some value here and there, e.g., show some continuity results right away and have some nice connections with volumes, but they are tricky to explain and CAN be omitted!!!

Uh, there is an easier proof of the Schwartz inequality based on Bessel's inequality. Since they did enough with orthogonality to do Bessel's inequality, they could have used that approach to the Schwartz inequality -- I first saw in P. Halmos.

They didn't make clear the close connections among inner products, covariance, and correlation -- maybe some readers will see those connections from what is in the slides.

They did the QT decomposition -- nice -- that is, for square matrix A, we can write A = QT where Q is orthogonal and T is triangular. They used that to solve systems of linear equations but omitted Gauss elimination and the associated approaches to numerical stability. For the Q, they emphasized the Gram-Schmidt process but neglected to mention that it's numerically unstable -- no wonder since are commonly subtracting large numbers whose difference is small, the basic sin in numerical analysis.

Of course, the authors are EE profs. Then it is interesting that another theme in the slides is getting close to much of the work in what computer science calls machine learning. E.g., their few slides on using classification to recognize digits 0-9 in handwriting is really cute, especially the graph that shows the sizes of the coefficients on top of the square that has the input data of hand written digits so that see which parts of the input data are the most relevant to the calculation. Cute.

Of course, there's much more to those fields that they omitted than included, but that's true also for even the best 5 star hotel luncheon buffet!!!

More fun stuff at

https://news.ycombinator.com/item?id=18648999