Timeline for Linear Algebra Texts?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 4, 2019 at 21:30 | history | edited | kjetil b halvorsen | CC BY-SA 4.0 |
spelling layout
|
| Jan 9, 2017 at 6:10 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
corrected minor typo (the question has been bumped anyway)
|
| Jun 20, 2010 at 1:06 | comment | added | Victor Protsak | Harry, don't get all worked up. Let me repeat the key point: $\textit{It's a harmful fallacy that conceptual understanding and applications are mutually exclusive.}$ There isn't any application in sight in your quotes, just comparisons between different formalisms. And for what it's worth, the last one is perfectly in line with Arnold's philosophy, while it doesn't conform well to Bourbaki's way of thinking. | |
| Jun 19, 2010 at 10:43 | comment | added | Harry Gindi | Artin's view here is very much the view of Dieudonné (as expressed in his book on linear algebra). I think that Arnold simplifies the world into black and white and attacks a straw boogeyman named Bourbaki. Hoffman and Kunze gives a very nice account of these geometric aspects as well as the algebraic ones. | |
| Jun 19, 2010 at 10:37 | comment | added | Harry Gindi | Then one proves all the customary theorems without ever talking of matrices and asks the question: Suppose we can find a basis of V which consists of eigenvectors; what does this imply for the geometric description of f? Well, the space is stretched in the various directions of the basis by factors which are the eigenvalues. Only then does one ask what this means for the description of f by a matrix in terms of this basis. We have obviously the diagonal form." - [Artin p. 13-14] | |
| Jun 19, 2010 at 10:36 | comment | added | Harry Gindi | Instead one should argue like this: Given a linear transformation f of the space $V$ into itself. Does there exist a line which is kept fixed by $f$? In order to include the eigenvalue $0$ one should then modify the question by asking whether a line is mapped into itself. This means of course for a vector spanning the line that $$f(X)=\lambda X.$$ Having thus motivated the problem, the matrix A describing f will enter only for a moment for the actual computation of X. It should disappear again. | |
| Jun 19, 2010 at 10:31 | comment | added | Harry Gindi | Matrix method: A product of a matrix $A$ and a vector $X$ (which is then an n-tuple of numbers) is defined; it is also a vector. Now the poor student has to swallow the following definition: A vector X is called an eigenvector if a number $\lambda$ exists such that $$AX=\lambda X.$$ Going through the formalism, the characteristic equation, one then ends up with theorems like: If a matrix A has n distinct eigenvalues, then a matrix $D$ can be found such that $DAD^{-1}$ is a diagonal matrix. The student will of course learn all this since he will fail the course if he does not. | |
| Jun 19, 2010 at 10:25 | comment | added | Harry Gindi | "Mathematical education is still suffering from the enthusiams which the discovery of this isomorphism has aroused. The result has been that geometry was eliminated and replaced by computations. Instead of the intuitive maps of a space preserving addition and multiplication by scalars (these maps have an immediate geometric meaning), matrices have been introduced. From the innumerable absurdities -from a pedagogical point of view - let me point out one example and contrast it with the direct description: | |
| Jun 19, 2010 at 10:23 | comment | added | Harry Gindi | @Victor: I don't agree one bit. I realize that Arnold's diatribe is very famous and loved by Bourbaki-haters everywhere, but I counter it with a famouse paragraph from Emil Artin's monograph "Geometric Algebra": | |
| May 23, 2010 at 1:16 | comment | added | Victor Protsak | I've had zero experience with any of the books named, but I completely support the point that Andrew made: abstract linear algebra strips the motivation and skips perhaps the most interesting part: applications. V.I.Arnold was very caustic about this "criminal Bourbakization and algebraization of mathematics". It's a harmful fallacy that conceptual understanding and applications are mutually exclusive. | |
| Mar 10, 2010 at 18:49 | comment | added | Harry Gindi | I find "applications" largely misleading in Linear algebra. The fact that linear algebra has been split up from a one semester course to a two semester course (matrix algebra and linear algebra) at many schools is a real shame. If you've read Emil Artin's "Geometric Algebra" monograph, you can see that it's very easy to take the isomorphism between an abstract finitely generated vector space and k^n for granted. Conceptual understanding is much more important. | |
| Mar 10, 2010 at 18:14 | history | answered | The Mathemagician | CC BY-SA 2.5 |