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> Isn‘t there anybody close to the Feynman of Linear Algebra? No. The subject is too young (the first book dedicated to Linear Algebra was written in 1942). Si

• rq1 3y ago
  Why should it buy you something is the real question. You don't need to understand it the way the "initial" author thought about it, should that person had given it more thoughts... History of maths is really interesting but it's not to be confused with math. Concepts are not useful as you think about them in economic opportunity case. Think about them as "did you notice that property" and then you start doing math, by playing with these concepts. Otherwise you'll be tied to someones way of thinking instead of hacking into it.
  • resource0x 3y ago
    Your post represents a common viewpoint, but I don't agree with it. I'm a retired programmer trying to learn algebra for the purposes of education only. I am not supposed to take an exam or use the material in any material way, so to speak. I'd like to understand. Without understanding motivations and (on the opposite end) applications I simply lose interest. I happen to have a degree in math, and I know for the fact that when you know (or can reconstruct) the untuition behind the theory - it makes a world of a difference. If this kind of understanding is not a goal, then what is? BTW, by "buying" I din't mean that it should buy me a dinner, but at least it's supposed to tell me something conceptually important within the theory itself. Example: in the LADR book, the chapter on dual spaces has no consequences, and the author even encourages the reader to skip it :).
  • tnecniv 3y ago
    I know more math than the average bear, but I think the parent has a point even if I don’t totally agree with them. Take for instance the dual space example. The definition of it to someone who hasn’t been exposed to a lot of math seems fine but not interesting without motivation — it looks just another vector space that’s the same as the original vector space if we’re working in finite dimensions. However, the distinction starts to get interesting when you provide useful examples of dual spaces. For example, if your vector space is interpreted as functions (for the novice, even they can see that a vector can be interpreted as a function that maps an index to a value), then the dual space is a measure — a weighting of the inputs of those functions. Even if they are just finite lists of numbers in this simple setting, it’s clear that they represent different objects and you can use that when modeling. How those differences really manifest can be explored in a later course, but a few bits of motivation as to “why” can go a long way. Mathematicians don’t really care about that stuff — at least the pure mathematicians who write these books and teach these classes — because they are pure mathematicians. However, the folks taking these classes aren’t going to all grow up and be pure mathematicians, and even if they are, an interesting / useful property or abstraction is a lot more compelling than one that just happens to be there.
    • rq1 3y ago
      There can be several motivations. Would it be more interesting to present these with the Gelfand triple instance? Does it have “more” to say than the initial raw definition? The concept can be used in different contexts and that’s what makes algebra being algebra. People have different motivations and usually that’s what brings new light into a field.
• Koshkin 3y ago
  > No Gilbert Strang (already mentioned by fellow commenters). > The subject is too young "The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged." (from Wikipedia)
  • resource0x 3y ago
    The first book was written in 1942 - it's mentioned explicitly in LADR. It doesn't mean the concepts didn't exist - they did, Frobenius even built a brilliant theory around them (representation theory), but the subject was defined quite loosely - apparently no one cared to collect the results in one place. It doesn't even matter much: I remember taking the course in 1974, and it was totally different from what is being taught today.
• ravi-delia 3y ago
  What? Linear Algebra is easily one of the best understood fields of mathematics. Maybe elementary number theory has it beat, but the concepts that drive useful higher level number theory aren't nearly so clear or direct as those driving linear algebra. It's used as a lingua franca between all sorts of different subjects because mathematicians of all stripes share an understanding of what it's about. From what you said there, it seems like you tried to approach linear algebra from nearly random directions- and often from the end rather than the beginning. If you're in it for the computation, Axler definitely isn't for you. There are texts specifically on numeric programming- they'll jump straight to the real world use. If you want to understand it from a pure math perspective, I'd recommend taking a step back and tackle a textbook of your choosing in order. The definition of a dual space makes a lot more sense once you have a vector space down.
  • ajkjk 3y ago
    I sympathize with the person you're responding to a lot more than you. It's very easy to understand what a dual space is. It's very hard to understand why you should care. Many of the constructions that use it seem arbitrary: if finite vector spaces are isomorphic to their duals, why bother caring about the distinction? There are answers to this question, but you get them somewhere between 1 and 5 years later. It is a pedagogical nightmare. Every concept should have both a definition and a clear reason to believe you should bother caring about it, such as a problem with the theory that is solved by the introduction of that concept. Without the motivating examples, definitions are pointless (except, apparently, to a certain breed of mathematicians). I've read something like 100 math textbooks at this point. I would rate their pedagogical quality between an F and a D+ at best. I have never read a good math textbook. I don't know what it is, but mathematicians are determined to make the subject awful for everybody who doesn't think the way they do. (I hope someday to prove that it's possible to write a good math textbook by doing it, but I'm a long way away from that goal.)
    • ravi-delia 3y ago
      I absolutely see what you're saying with that. I think I'm definitely the target audience of the abstracted definition, but I've long held that every new object should be introduced with 3 examples and 3 counter-examples. But you said it yourself- that's the style pure math texts are written in! Saying that "we" as a species don't have a good understanding of linear algebra is unbelievable nonsense. I can't conceive of the thought process it would take to say that with a straight face. The fact is, 10 separate YouTube lectures disconnected from anything else is just the wrong way to try and learn a math topic. That's going to have as much or more to do with why dual spaces seem unmotivated as the style of pedagogy does.
• ndriscoll 3y ago
  > Why should I care about different forms of matrix decomposition? What do they buy me? A natural line of questioning to go down once you're acquainted with linear maps/matrices is "which functions are linear"/"what sorts of things are linear functions capable of doing?" It's easy to show dot products are linear, and not too hard to show (in finite dimensions) that all linear functions that output a scalar are dot products. And these things form a vector space themselves, the "dual space" (because each element is a dot-product mirror of some vector from the original space). So linear functions from F^n -> F^1 are easy enough to understand. What about F^n -> F^m? There's rotations, scaling, projections, permutations of the basis, etc. What else is possible? A structure/decomposition theorem tells you what is possible. For example, the Jordan Canonical Form tells you that with the right choice of basis (i.e. coordinates), matrices all look like a group of independent "blocks" of fairly simple upper triangle matrices that operate on their own subspaces. Polar decomposition says that just like complex numbers can be written in polar form re^it, where multiplication scales by r and rotates by t, so can linear maps be written as a higher dimensional multiplication/scaling and orthogonal transformation/"rotation". The SVD says that given the correct choice of basis for the source and image, linear maps all look like multiplication on independent subspaces. The coordinate change for SVD is orthogonal, so another interpretation is that roughly speaking, SVD says all linear maps are a rotation, scaling, and another rotation. The singular vectors tell you how space rotates and the singular values tell you how it stretches. So the name of the game becomes to figure out how to pick good coordinates and track coordinate changes, and once you do this, linear maps become relatively easy to understand. Dual spaces come up as a technical thing when solving PDEs for example. You look for "distributional" solutions, which are dual vectors (considering some vector space of functions). In that context people talk about "integrating a distribution with test functions", which is the same thing as saying distributions are dot products (integration defines a dot product) aka dual vectors. There's some technical difficulties here though because now space is infinite dimensional, and not all dual vectors are dot products, e.g. the Dirac delta distribution delta(f) = f(0) can't be written as a dot product <g,f> for any g, but it is a limit of dot products (e.g. with taller/thinner gaussians). One might ask whether all dual vectors are limits of dot products and whether all limits of dual vectors are dual vectors (as limits are important when solving differential equations). The dual space concept helps you phrase your questions. They also come up a lot in differential geometry. The fundamental theorem of calculus/Stokes theorem more-or-less says that differentiation is the adjoint/dual to the map that sends a space to its boundary. I don't know off the top of my head of more "elementary" examples. It's been like 10 years since I've thought about "real" engineering, but roughly speaking, dual vectors model measurements of linear systems, so one might be interested in studying the space of possible systems (which, as in the previous paragraph, might satisfy some linear differential equations). My understanding is that quantum physics uses a dual space as the state space and the second dual as the space of measurements, which again seems like a fairly technical point that you get into with infinite dimensions. Note that there's another factoring theorem called the first isomorphism theorem that applies to a variety of structures (e.g. sets, vector spaces, groups, rings, modules) that says that structure-preserving functions can be factored into a quotient (a sort of projection) followed by an isomorphism followed by an injection. The quotient and injection are boring; they just collapse your kernel to zero without changing anything else, and embed your image into a larger space. So the interesting things to study to "understand" linear maps are isomorphisms, i.e. invertible (square) matrices. Another way to say this is that every rectangular matrix has a square matrix at its heart that's the real meat.