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My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multip
by SonOfLilit 9y ago
My test for linear algebra books is how they first present matrices and matrix multiplication.
If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multiplication of their matrix forms, I throw the book away in disgust. I throw most books away in disgust.
This is the second book I've seen that does it right, but unlike the other one [1] this one wraps the very correct presentation of matrices in so much technical language and such a boring cover story that I almost threw it away in disgust anyway.
My greatest wish in STEM education is that we teach linear algebra better. It's such low hanging fruit and could change so much!
[1] which I saw linked in HN a year ago and don't remember the name of, but it was something like "linear algebra taught the correct way" and was apparently well known in the States so ask your friends
- jpau 9y agoMight [1] be "Linear Algebra Done Right" by Axler? http://www.springer.com/gp/book/9783319110790 http://www.springer.com/gp/book/9783319110790
- andars 9y agoThe Friedberg, Insel, Spence book [1] passes your test. Linear transformations first, then matrices as a representation of transformations with respect to a specified basis. Depending on your tastes, you may find it too dry, but imo it is a clean presentation. [1]: https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514 https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/...
- abecedarius 9y agoAnother text that starts with linear transformations is Apostol’s Calculus. It’s somewhat more concrete than Axler, IIRC — e.g. it doesn’t try to avoid determinants, but starts with properties of volume-scaling and then develops the determinant formula out of those. We used this text for freshman/sophomore math back in the 80s, so my memory is fuzzy; but I liked it a lot.
- begriffs 9y agoI think the book you're talking about is Axler's "Linear Algebra Done Right." I worked through the problems in this book with some friends, and it is a very good presentation. After doing that book I wanted to see what other books were out there, which ones were the good ones, and sort of classify them. I came up with this list: https://begriffs.com/posts/2016-07-24-best-linear-algebra-books.html https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo... I classify the books as Generalist (like Axler's), Theoretical (starting from e.g. modules rather than vector spaces), Numerical (matrix normal forms and algorithms), Practice (books full of problems), and a few other categories.
- linkmotif 9y agoAweomse. Thank you. Jealous of you and your friends. Wish I had a Pythagorean squad :)
- wolfgke 9y ago> After doing that book I wanted to see what other books were out there, which ones were the good ones, and sort of classify them. I came up with this list: https://begriffs.com/posts/2016-07-24-best-linear-algebra-books.html https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo... Ironically you put "Jänich, K. (1994). Linear algebra. New York: Springer-Verlag." into the "Theoretical" section: In Germany this book (its German original) is not that well-regarded, since it is considered as far to shallow. Much better (and more hard to read) German textbooks are Gerd Fischer - Lineare Algebra: Eine Einführung für Studienanfänger (Linear Algebra: An Introduction for freshmen) Siegfried Bosch - Lineare Algebra (and its companion book: Siegfried Bosch - Algebra)
- begriffs 9y agoYou're right, I made a mistake. Just reviewed the table of contents on Amazon and it looks pretty short, and not particularly theoretical. Also poorly reviewed. I don't remember how it got on my list but I'm removing it.
- sn9 9y agoI've heard Shilov's text described as a great complement to Axler's.
- j2kun 9y agoTo this book's credit, it covers a LOT more material than Axler's book. And it's more heavily focused on optimization methods for which the details of tabular representations of matrices takes center stage.
- tomjakubowski 9y agoStrang’s textbook (and lectures) introduce matrix multiplication AB as combinations of A’s columns. He uses the cell-by-cell / “dot product” formulas just to check his work, which seems reasonable to me. I’m a big fan of “combinations of columns”. I sometimes even hear that in Strang’s voice when I’m doing some matrix multiplications. Axler was way too abstract for me as a first linear algebra book. I look forward to returning to it once I finish Strang.
- internetman55 9y agoStrang's exposition can make you a wizard of matrix multiplication and give you quite a good intuition for various properties of matrix operations, in my opinion.
- spraak 9y agoI'm curious what you think of the "No Bullshit Guide to Linear Algebra" [1]? I'm considering buying it to refresh my knowledge from school. Or what books do you suggest? [1] https://www.amazon.com/No-bullshit-guide-linear-algebra/dp/0992001021 https://www.amazon.com/No-bullshit-guide-linear-algebra/dp/0...
- ivan_ah 9y agoHere is a direct link to the page in the preview of the book where the matrix-product-is-the-same-as-linear-transformation idea first appears: https://minireference.com/static/excerpts/noBSguide2LA_preview.pdf#page=53 https://minireference.com/static/excerpts/noBSguide2LA_previ... I can't quite tell if the book passes the test, since the entry-point is the definition of matrix-vector product which is very close to a "complicated formula with a couple of nested sigmas" but the book also mentions the notion of matrix representations, so hard to tell overall.
- SonOfLilit 9y agoIt... more or less passes the test? I mean, it defines matrix-vector product in a nice but abstract way, and then in the next paragraph explains why we chose that definition. And it does say "this, this here is the one important idea in this book", which it gets many points for. I really would prefer a textbook to start with "ok, here's something we want to do. Lets figure out a formula for it. Now lets give it the name matrix-vector product". Same for matrix-matrix products. I don't know any really good linear algebra books, in my school it was taught the worst possible way (define a field, rote-learn the mechanical operation of matrix multiplication, talk about vector spaces for a while, talk about matrices, first mention linear transformations). Some people on this thread gave other suggestions, I'd start reading a few and choose the one I connect with the best. Anyway, I really learned linear algebra from using it.
- deleted 9y ago[deleted]
- wging 9y agoIIRC, Apostol's Calculus, volume 2, does it right (linear transformations first, then matrices).
- jacobolus 9y agoStarting with an abstract formal description of vector spaces and linear transformations is not a pedagogically useful introduction, and it leaves out a huge amount of the historical/motivational context explaining most of the conventions used in linear algebra, and even the mental/conceptual understanding most working mathematicians have about the meaning of linear models and their use in various parts of mathematics (not even to mention people in other STEM fields). The two places to start historically (and easily accessible to high school students) are: (1) understanding and working with displacement vectors in 3-dimensional Euclidean affine space and in general thinking about transformation geometry (sometime later this can be extended to other kinds of non-Euclidean or non-metrical geometry), especially with reference to problems in Newtonian mechanics. (2) systems of linear equations: this one is inherently coordinate-heavy and matrix based, at least to start out, and explains our conventions for how matrices are written, index order, multiplication of matrices by "column" vectors on the right, the use of matrix equations to fold several equals signs into one, etc. After that I'd call out the manipulation of vectors of polynomial coefficients as an accessible additional concrete example of a linear space. Discussion of other linear spaces or more purely abstract treatments proving properties from axioms can come sometime later, after students have familiarity with some of those tools, and after they have applied them to some problems in statistics, multivariable calculus, computer graphics, ODEs, optimization, etc.
- jacobolus 9y agoAnd to follow-up, note that this book in particular is for a graduate-level course in the department of computer and information science, which has an undergraduate-level linear algebra course as a prerequisite.
- nextos 9y agoDo you have any other tests for calculus, analysis, abstract algebra, discrete mathematics, probability or statistics books?
- kccqzy 9y agoFor me my test for multivariable calculus text is their treatment of the chain rule. If the book says something like “derivative of a composition is the composition of the derivatives” it’s good. If they instead say something involving a sigma and things like ∂f/∂x ∂x/∂t it’s likely bad.
- apricot 9y agoWhat do you mean by "derivative of a composition is the composition of the derivatives"?
- kccqzy 9y agoYou need to think of a derivative as just a function that takes a function, a point, and returns a linearized function that best approximates the original function at that point. So basically the chain rule states that if you have two functions F, G composed together and you want to find the derivative (the linearized approximation), you simply compute the linearized function for both F and G, and compose the linearized version afterwards. You can read the Wikipedia article on the chain rule and it eventually gives this simple and elegant formula for the chain rule: https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0f133ce31911e43c764654f1413b98c911dc7b https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0f... But the unfortunate reality is that too many textbooks formulate the chain rule in such a complicated manner that it obscures the simplicity and elegance of the chain rule.
- apricot 9y agoThank you for the explanations. I agree with you that this more abstract view of the chain rule (and the derivative itself) is superior to the sum-of-products formula one usually sees in a first course in multivariable calculus, but I feel most students have to learn the complicated, technical version first before they can see the beauty of the more abstract one.
- catnaroek 9y ago> My greatest wish in STEM education is that we teach linear algebra better. That's easy. Vector spaces and linear transformations are best understood abstractly. Matrices are best understood as visually and computationally convenient representations of tensor products, so don't mention matrices at all until well after you have established the basic properties of tensor products, in particular, the natural isomorphism between `L(V,W)` and `Dual(V) (x) W`.
- kccqzy 9y agoI don’t necessarily agree though. I’ve seen books first trying to introduce linear transformation without matrices and frankly to a beginner it’s even more confusing to be told f is linear if f(ax+y)=af(x)+f(y). Some students really need a concrete grasp of matrices with actual numbers they are familiar with before going into the abstract definition. I’ve seen worse though: those that attempt to shoehorn abstract algebra in the process by first rigorously defining a field. And no, teaching multiplication doesn’t involve memorizing a formula; it’s a simple mechanical process of arranging one matrix on the left the other at the top and multiplying/adding their corresponding rows and columns. Once this process is familiar to a student, they will have no trouble writing out the formula with nested sigmas.
- jknz 9y agoI would teach linear algebra with a book that barely mentions linear transformations. In finite dimensions, linear transformations and matrices are exactly the same object mathematical objects, with very different notations (matrix notation (boxes with numbers inside) vs the linear space/linear transformation notation). I would rather the students to learn deeper mathematics only in matrix notation, rather than to master less substantial mathematics with both notations. Teaching both notations may reinforce the idea that matrices and linear transformations are different mathematical objects in finite dimension. Teaching the more abstract notation is mainly useful in infinite dimension (Hilbert spaces).
- btilly 9y agoThis is exactly backwards. Whether you're discussing the Jacobian of a function, or change of basis matrices, learning the matrix formula is a lot less useful than seeing how it falls out of the linear function definition. The formula is hard to memorize and gives no intuition for why anything is true. But from the linear function definition it is easy to reconstruct the formula. In fact this is so true that I would say that anyone who only knows the matrix definition does not actually understand linear algebra.
- catnaroek 9y agoAnd you would forever cripple your students. As potential users of linear algebra, they would be mentally mutilated beyond all hope of regeneration.
- jacobolus 9y agoYou have succumbed to the coordinate virus, http://geocalc.clas.asu.edu/pdf/MathViruses.pdf http://geocalc.clas.asu.edu/pdf/MathViruses.pdf
- wolfgke 9y ago> In finite dimensions, linear transformations and matrices are exactly the same object mathematical objects They are not: Matrices represent linear transformations with respect to a given basis. Linear transformations are completely independent from any chosen basis.
- internetman55 9y agoI don't understand what's so wrong about that. Surely the interested student would like to learn about the properties of matrix arithmetic as well as vector spaces, linear transformations and their representation as matrices, etc. I've had a quite accomplished mathematician for a professor who was very adamant that students should learn about basic matrix operations before the abstract theory. According to him, his undergraduate education at Princeton did things the "right" way, but left him quite confused about the main ideas and motivations. He ended up making some quite important contributions to the theory of representations of some certain algebraic structures if I recall correctly.
- SonOfLilit 9y agoThey way I was taught was like learning how to do long-form multiplication without mentioning that "the product of 5 and 6 is 5+5+5... 6 times".
- igniousrock 9y agoTo me, matrices and matrix multiplication really represent the "calculus" of linear algebra. When you are studying real analysis, the theory part (closed, open sets, definition of limit points, continuity, etc) are not actually that helpful when doing a computation (calculating and integral or derivative). It often makes sense to teach a student the calculation side before the theory because its easier to get a intuition for.
- Tomte 9y agoStupid question: is the order of elements in matrix notation just convention or is there a deeper reason for it? Could we just as well have used „transposed matrices“?