Timeline for Linear Algebra Texts?
Current License: CC BY-SA 3.0
33 events
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| Jan 9, 2017 at 6:09 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
corrected minor typo (the question has been bumped anyway)
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| Jun 19, 2010 at 14:01 | comment | added | Steven Gubkin | The ONLY way I ever remember the definition of matrix multiplication is by remembering that the $i^{th}$ column tells you where the $i^{th}$ basis vector goes. To write the composite down, I think about where my basis vectors are going when I compose the two indicated linear transformations. Learning to multiply matrices without this understanding is truly pointless. | |
| May 23, 2010 at 7:29 | comment | added | Harry Gindi | I don't understand how learning to compose linear maps is pointless without multiplying matrices? It's pointless if you intend to compute something, but pretty much every theorem in linear algebra can be proven without resorting to explicit matrix computation, and even if you did need to use matrices, the definition of matrix multiplication $(AB)_ij=\sum_{k=1}^n A_{ik}B_{kj}$ should suffice. | |
| May 23, 2010 at 6:49 | comment | added | Victor Protsak | @Harry: Bear in mind that what works for $\textit{you}$ (and other math majors at UM), doesn't necessarily work for others. And you have just confirmed that you were comfortable with abstract algebra, at the level higher than most non-majors ever see, before starting out. Learning composition of morphisms without being able to multiply matrices is $\textit{truly}$ pointless. One imperfect analogy: it's possible to learn AG from EGA or Harstshorne ("it has been done"), but as first books they are nowhere near Shafarevich, Cox-Little-O'Shea, Reid, Mumford, and any number of other texts. | |
| May 23, 2010 at 3:20 | comment | added | Harry Gindi | Learning to do matrix computations without understanding that matrices represent linear transformations and that matrix multiplication represents composition of morphisms is rather pointless. This is why Hoffman and Kunze do not emphasize them early on. Endless matrix computations also ruin the students' perceptions of the whole subject. This is why I favor the abstract-first approach, because otherwise, the whole subject makes no sense. | |
| May 23, 2010 at 3:15 | comment | added | Harry Gindi | @Victor: We used this book in my second semester of mathematics right after our first semester doing introductory real analysis/calculus. The only algebra experience we had was from our first semester homework, where we learned the axioms of group theory and proved such theorems as: the first isomorphism theorem, the orbit-stabilizer theorem, and even the second isomorphism theorem. I don't understand why people say that it can't be done when it has been done. This book has been the standard linear algebra book for 30 years. How is it suddenly now too hard? | |
| May 23, 2010 at 1:31 | comment | added | Victor Protsak | From the description given (starts with a $\textit{review}$ of matrices and fields, uses modules and exterior algebra, etc), it is patently obvious that this book (HK) is unsuitable to people without abstract algebra under their belt. Quite a bit of mathematical maturity that you cannot reasonably expect from non-math majors with or without prior proof experience is required as well. | |
| Mar 10, 2010 at 22:21 | comment | added | Harry Gindi | Yes, I meant determinant. | |
| Mar 10, 2010 at 22:00 | comment | added | Steven Gubkin | The volume description makes it pretty clear that it is invariant under change of basis. I must admit that I have not learned too much about the exterior algebra point of view, and Hoffman and Kunze seems like a nice place to pick this up! I am putting this book on my list for the summer. Also, I think you meant to say determinant, not derivative? | |
| Mar 10, 2010 at 21:34 | comment | added | Harry Gindi | We can derive the standard formula for the derivative explicitly by the definition of the nth exterior power literally as soon as we choose a basis (this is literally immediate). It is this naturality that explains why the derivative is invariant under change of basis. | |
| Mar 10, 2010 at 21:32 | comment | added | Harry Gindi | The definition I gave using the exterior algebra motivates the form that darij is talking about. The "unique top-level alternating multilinear form" is true as a statement about the row space or column space of the matrix, but this is not the intrinsic definition of the determinant. No, the determinant is the unique induced map determined by the functoriality of the exterior product. That is, $Det:Hom(V,W)\to Hom(\wedge^n(V),\wedge^n(W)$. We can also say that the determinant is a natural transformation between the identity and the top exterior power functor. | |
| Mar 10, 2010 at 20:49 | comment | added | darij grinberg | ... revealed to have a nice explicit form, will believe that you are just trying to make their life harder. | |
| Mar 10, 2010 at 20:48 | comment | added | darij grinberg | "how do you justify the general formula": Well, it's a logical generalization of the 2x2 and 3x3 formulas. Nothing speaks against the characterization of the det as the unique alternating multilinear form; however, I don't like this characterization as the definition of the det. Students in their first year usually do not yet grasp the subtle difference between "definition that is hard to use, but was put in the beginning for tactical purposes" and "explicit definition that one can use for computations", and when presented with an object that is defined implicitely and only later ... | |
| Mar 10, 2010 at 20:44 | comment | added | Steven Gubkin | In your examples based approach, how do you justify the general formula? I am personally inclined towards a combination of the geometric approach, and the characterization as the unique alternating multilinear form (up to rescaling). Volume obviously satisfies the multilinear form property, and it is alternating if you pick sign conventions correctly, so these approaches support each other. Then you can ask, when will the image of a vector space have lower dimension? When the volume of the image is 0 of course! This then ties is it back to solving equations. | |
| Mar 10, 2010 at 20:40 | comment | added | darij grinberg | By the way, do you know what is funny? "Prove that if A and B are square matrices of the same size and AB = I, then BA = I." This exercise is in Axler's Chapter 10, so I assume he wants the reader to use determinants here. Needless to say, this is one of the cases where the proof without determinants is easier (though the one with determinants works over any commutative ring, which is also nice but never valued by Axler, who doesn't even show applications of linear algebra over F_2 to combinatorics and coding). | |
| Mar 10, 2010 at 20:38 | comment | added | darij grinberg | ... advanced number theory, when you show that the sum and the product of two algebraic integers is an algebraic integer; basically, this proof tells you how to work with algebraic numbers with 100% precision, rather than approximating them as complex numbers and hoping that your precision was enough)!. | |
| Mar 10, 2010 at 20:36 | comment | added | darij grinberg | ... the eigenvalues as its roots, but it's the algebraic definition which shows trivially that every coefficient of this polynomial is continuous and differentiable in A (this isn't quite clear from eigenvalues), exactly computable (the world isn't all about floating-point computations), integer if A is an integer matrix, real if A is a real matrix (this is easy using eigenvalues, but it's absolutely trivial using the det (1 - AT) approach), etc. Besides, Axler completely avoids the classical adjoint, which he doesn't need but is often used (for instance, in the very beginning of ... | |
| Mar 10, 2010 at 20:33 | comment | added | darij grinberg | ... through a global (determinant as a function on the space of all matrices) and implicit definition. I think the best way is the good old "do Gaussian elimination for a symbolic system of equation, and see the pattern" way. Of course, one should then show the relation to area and volume in R^2 and R^3, the whole multilinearity properties and the criteria for linear independency. As for the characteristic polynomial, one should define is as det (1 - AT), keeping in mind that in the case of MOST matrices (those without repeated eigenvalues) this is the same as the polynomial with ... | |
| Mar 10, 2010 at 20:30 | comment | added | darij grinberg | He does, in the 10th Chapter. But I fear that most students learning linear algebra by this book will see these facts about determinants as mere sidenotes and the definition through triangularization as the main property of the determinant. Note that I also disagree with the definition of the det as the unique alternating form such that ... (you know what I mean); while mathematically correct, it is pedagogically unsatisfactory, as introducing a local and computable concept (local since there is the determinant of one given matrix, and computable since there is a formula) ... | |
| Mar 10, 2010 at 20:19 | comment | added | Steven Gubkin | Axler does include determinants at the end of the book as how much the volume of the unit cube gets scaled by a linear transformation, and then applies this to some multivariable calculus. This doesn't seem like a bad intro. | |
| Mar 10, 2010 at 20:17 | comment | added | Steven Gubkin | In an Axler course it would be very easy to take one week out of the course to motivate determinants and learn how to compute with them. I do not see why this is an argument against the entire book. The book is really about normed linear spaces, and so the fact that he doesn't work with more general coefficients is not surprising. Avoiding determinants really does make many of the proofs in his book more conceptual than the "standard proofs". But I guess "more conceptual" is really just an aesthetic thing. | |
| Mar 10, 2010 at 19:44 | comment | added | Harry Gindi | I laughed out loud at "Jordanize this 5x5 matrix". | |
| Mar 10, 2010 at 19:35 | comment | added | darij grinberg | ... Axler's course, just as any other, offers more than enough occasions for a bad lecturer to spam students with stupid computation exercises like "compute the eigenvalues" and "jordanize this 5x5 matrix". Both an Axlerian course and a totally algebraic one can offer quite a lot of motivation, but of course if a lecturer doesn't want to motivate, he won't. | |
| Mar 10, 2010 at 19:32 | comment | added | darij grinberg | A linear algebra course should be there for everyone, not just for the future functional analysts. In differential geometry, you will have to derive the determinant of matrix; good luck doing it if you just know the determinant as a coefficient of the characteristic polynomial, which is defined through triangulation of the matrix. -- I think the difference between Axler's take on linear algebra and the algebraic approach has not much to do with computations. ... | |
| Mar 10, 2010 at 19:21 | comment | added | Steven Gubkin | Maybe not everyone is going into algebraic geometry? I think that a lot of the proofs in Axler suggest the right way to think about things in functional analysis. Maybe it was just that my first linear algebra course was totally computational and had very little motivation, and my second exposure to linear algebra out of Axler showed me that there were actually some beautiful ideas there. | |
| Mar 10, 2010 at 19:00 | comment | added | Harry Gindi | Even better and more algebraic, the determinant emerges as the one-dimensional linear map (hence an element of the field) induced on the top-level exterior powers. This is the correct conceptual explanation (and is included in Hoffman and Kunze). | |
| Mar 10, 2010 at 18:53 | comment | added | darij grinberg | For someone who plans to work in algebra or algebraic geometry, linear algebra learnt from Axler is mostly wasted time. I don't understand what he has against the notion of determinant; this notion (with the sum-over-permutations formula that he seems to hate) comes out straightforwardly if one tries to apply Gaussian elimination to a general systems where the coefficients of the system are variables. | |
| Mar 10, 2010 at 18:50 | comment | added | darij grinberg | The problem with Axler is that he tries to avoid algebra at all cost - when it makes things easier and when it makes thing harder. Sometimes much harder, actually. Even if one goes into physics and never has to work over a ring different from R and C, one will realize one day that in order to compute the characteristic polynomial one doesn't have to bring the matrix in upper triangular form (no joke, this is how Axler defines the characteristic polynomial), and that often, the characteristic polynomial matters and upper diagonalization doesn't. | |
| Mar 10, 2010 at 18:25 | comment | added | Steven Gubkin | Have you actually looked at Axler? You do get a lot of hands on experience computing using bases. He does not avoid matrices in any way - he just makes it very clear that the only thing you care about them is that the ith column vector tells you where the ith basis vector goes. I cannot think of a book where the standard proofs from linear algebra are made more conceptual than Axler's. If you really want to work with determinants, then it would not be too hard to integrate these into an Axler based course. | |
| Mar 4, 2010 at 14:41 | comment | added | Mark Meckes | I also first learned linear algebra from H&K and share fpqc's enthusiasm for it, but I don't think it counts as "relatively gentle". I think RH is right that H&K is pretty advanced in spirit if your students don't already have experience with rigorous proof-based math. Having taught from Axler, which does have many nice points, I can also confirm the problems with it that fpqc describes. | |
| Mar 4, 2010 at 3:33 | comment | added | Dan Ramras | That's where I first learned linear algebra (not counting the material in multivariable calc). I liked it a lot. Not sure how suitable it would be for my course. I'll have to look back at it. Thanks for the warning regarding Axler. | |
| Mar 3, 2010 at 19:26 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 227 characters in body; added 553 characters in body
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| Mar 3, 2010 at 19:21 | history | answered | Harry Gindi | CC BY-SA 2.5 |