Makes sense. Something else I noticed, vector notation does not specify a coordinate system. The cartesian coordinate interpretation is a choice we make. Yes, the keyword here is 'basis'. You represent a vector by giving two pieces of data, 1 an ordered list of coordinates, and 2 a basis. The vector is then a linear combination of the basis elements, and the coordinates tell you how to form that linear combination. For example, let's use the standard Cartesian basis consisting of unit vectors e1, e2, e3 which point north, east, and up, informally speaking.
If the coordinates were given with respect to a different set of basis vectors, then you would take the linear combination using those vectors instead. Note the similarity of how a basis works to how a base system works representing numbers. Ok, I understand. But as used in computer languages, a vector can be simply 2 numbers. No coordinates or basis are implied. That's what I meant. Aha, yes. Computer languages borrowed the word 'vector', but they have basically nothing to do with the mathematical structure from linear algebra. It's best to keep them completely separate in your mind.
So in math, when we say "vector" coordinate system is a given, as you explained? If a coordinates are given, then they will be given with respect to a basis. Koshkin 6 months ago. Eventually the fog will dissolve by itself, and these kinds of questions will seem to you either naive or devoid any real substance, or just uninteresting compared to everything else that you have learned.
This advice does not apply to me. I don't want to learn mathematics. I'm more interested in learning parts of mathematics that interest me at the moment. And I think philosophy comes before mathematics. Or mathematics is the philosophy of quantities. Both philosopnies are based on definitions. A book on applied linear algebra with a focus on regression and no mention anywhere of the singular value decomposition??
IgniteTheSun 6 months ago.
This book has a lot of very interesting applications and seems to cover information not normally found in first books on Linear Algebra e. It doesn't, however, cover several topics normally covered in the first year of linear algebra e. As with most engineering books, there are no solutions provided. An excellent supplement to other Linear Algebra textbooks.
Given its focus on applications, will hold the interest of engineers and other technical folk but may not be loved by mathematicians who may prefer a more rigorous approach. Applied linear algebra is such a great idea. Linear algebra is relatively easy to understand and used everywhere. But the material is so damn boring since it's a lot of arthimetic. Even the homework problem is boring since there is no specific purpose. Typical LA courses in math departments have a bizarre focus on being able to do Gaussian elimination by hand and stuff like that. It's not particularly useful or even mathematically interesting.
LA courses would be so much more useful if they just stuck to theory and only had computer applications. I found knowing the mechanics of Gaussian elimination to be really helpful when learning about algorithms like SVD or using Householder transformations. Knowing how the matrices became triangularized gave me something to hang the new information on.
Just finished my intro linear algebra class yesterday. The class was a bit more abstract in nature, so some of the chapters in this look like they could be nice application oriented follow ups to it! How much overlap is there between this and Stanford's EE? Can someone in the know comment on the differences between what is covered?
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Great Resource! Ijust flipped some pages and saw it is great book.
Wished my school offered this. I went though the slides.
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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.
Applied Linear Algebra 1
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. 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. 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. 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.
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. 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. 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. 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 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. 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!!!
Looks like a nice paper! The paper says how to go beyond what is in Boyd, et al. For that material I would have been tempted just to use the old approach of determinants and the roots of the characteristic polynomial , the Hamilton-Cayley theoem, etc. Saved the paper! There is also the chapter in P. Halmos, Finite Dimensional Vector Spaces on multi-linear algebra which at the time I read it I took it as an abstract approach to determinants, maybe also a start on exterior algebra of differential forms, but maybe there's a long shot chance that that Halmos chapter is related to multi-dimensional determinants.
Can't read ALL the books on the shelves of the research libraries or even all the recent ones so have to be selective, to focus or as a startup entrepreneur before spending hundreds of hours in such a book hope the author got tenure ask "Why should I? I am sure Gelfand, Kapranov and Zelevinsky given their other math accomplishments all got tenure track positions when they emigrated.
Will give Halmos another look. Must be a great grand son or some such. The book is from and the research is from just before USSR fell. Proceed with caution.. Boyd is a great teacher! I highly recommend his course on linear dynamical systems  and convex optimization  too. The former is beginner stuff, and while convex optimization is more advanced, both are very engaging and clearly explained, with lots of anecdotes and practical examples! CamperBob2 6 months ago. Kind of inauspicious to see this kind of thing on page 5: A standard unit vector is a vector with all elements equal to zero, except one element which is equal to one Huh?
I've never heard the term "standard vector" before, and a "unit vector" is a vector whose magnitude is one. There is no requirement that one element be equal to one. Maybe that's why the download is free RossBencina 6 months ago. You could view it as inauspicious that the treatment doesn't begin with abstract vector spaces, but there is always Axler.
For what it's worth, I find it inauspicious that after taking three pure-math oriented Linear Algebra courses I never saw least squares nor the SVD.
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I'm looking forward to taking a look at Prof Boyd's book. Point being, their definition is just plain wrong. If that's how the authors describe a unit vector, I don't think this is the book you want to use to learn about SVD. Well, the way I see it, when you teach applied science you often want to sacrifice some rigor so that your students could focus on what was intended to be learned in the first place.
But there is the requirement that the other elements be equal to zero. It's right there in the sentence you quoted. True enough, thanks -- fixed. Still a completely bogus definition. What is your personal definition of the term "unit vector? Are there only N unit N-vectors, as the book says, or are there an infinite number of them?
My definition of unit vectors is the same as yours. However, the book does not say there are only N unit N-vectors. So why don't they tell readers what unit vectors actually are , in the general case? It's a rather important elementary concept, isn't it? Why bother describing a specific case using ambiguous language the parentheses and omit the one property that actually makes a unit vector a unit vector?
Bad writing in a math textbook is a weird thing to defend. Hacker News new past comments ask show jobs submit. FranzFerdiNaN 6 months ago This should be your first calculus book if you are learning from scratch. Connarhea 6 months ago I'm coming to the end of my first year 6 year part time Comp Sci course and have seen that we have options for AI and Machine Learning modules in future years. What is special about UC? Study Akoranga. Life at UC. Support services Tautoko. Support services. Finding support Need to talk?
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Bachelor's degrees. Undergraduate certificates and diplomas. Discretizing a differential equation over this cube creates a system of a million equations in a million unknowns. Linear systems also tend to be sparse, described by a matrix with mostly zero entries. In the example above, a solution every point in the cube depends on every other point in the cube, but not directly. The vast majority of this matrix is zeros since no page links to more than a few of the 4. Linear algebra at scale is not just a larger version of what you learn in high school.
Because matrices are so large and so sparse, even storing and accessing big sparse matrices requires some cleverness.