Timeline for Why do some linear algebra courses focus on matrices rather than linear maps?
Current License: CC BY-SA 4.0
22 events
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| Mar 18, 2021 at 0:38 | answer | added | dflora | timeline score: 1 | |
| May 17, 2020 at 23:19 | answer | added | James S. Cook | timeline score: 2 | |
| May 17, 2020 at 22:44 | answer | added | Jonny Evans | timeline score: 2 | |
| S May 17, 2020 at 19:18 | history | edited | amWhy | CC BY-SA 4.0 |
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| May 17, 2020 at 18:53 | review | Suggested edits | |||
| S May 17, 2020 at 19:18 | |||||
| May 17, 2020 at 0:20 | answer | added | guest | timeline score: 1 | |
| May 16, 2020 at 18:25 | answer | added | Daisuke Aramaki | timeline score: -1 | |
| May 16, 2020 at 10:26 | comment | added | Dan Fox | The matrices of an endomorphism with respect to different choices of ordered bases are similar, but there are contexts in which one works with matrices as such, rather than as representatives of their orbits under similarity. Such contexts are frequent in numerical linear algebra. For example, the condition number of a matrix with respect to inversion is $||A||||A^{-1}||$ and this is not a similarity invariant. | |
| May 16, 2020 at 4:01 | comment | added | Andrew T. | I guess this is the reason why there is applied mathematics in addition to pure mathematics? And also the broad range of quality between different universities/colleges across the world. | |
| May 15, 2020 at 21:36 | answer | added | Lars H | timeline score: 4 | |
| May 15, 2020 at 18:00 | comment | added | Dave L Renfro | spectral theorem for self-adjoint operators --- I suspect if this even makes sense for students in the course, then linear maps will probably be the primary focus (possibly after some introductory/motivational material on matrices and linear systems of equations). Anyway, I see your question as ill-posed, since there are many different levels of linear algebra. Indeed, from my personal experience at several U.S. universities, there tend to be three levels of linear algebra, which I've described in this answer. | |
| May 15, 2020 at 10:39 | comment | added | Dan Is Fiddling By Firelight | @NickC I'd take that argument a step farther. If you modify the course too far in favor of purity vs the sort of computations working engineers need at some point the school of engineering will modify its curriculum to drop MATH123 - Linear Algebra in favor of a newly created ENG456 - Useful Stuff From Linear Algebra course. | |
| May 15, 2020 at 8:12 | history | edited | Kostya_I | CC BY-SA 4.0 |
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| May 15, 2020 at 5:25 | history | became hot network question | |||
| May 15, 2020 at 4:06 | comment | added | user1815 | I would say that engineers (and applied mathematicians) need to study matrices more than most other mathematicians because they end up computing with matrices. When you compute with matrices (in floating point!), they don’t exactly behave like linear transformations. The matrix representation can matter. | |
| May 15, 2020 at 2:00 | answer | added | Nate Bade | timeline score: 23 | |
| May 15, 2020 at 1:56 | answer | added | Alexander Woo | timeline score: 9 | |
| May 15, 2020 at 1:55 | comment | added | Nick C | @XanderHenderson There's that quote from Irving Kaplansky, speaking of Paul Halmos: "We (Halmos and Kaplansky) share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury." So, not only are our beginning students aided by computation, but so were the greats. | |
| May 14, 2020 at 23:46 | comment | added | Xander Henderson♦ | Because, like lower division calculus, lower division linear algebra focuses on computation rather than concepts. One can compute with matrices. It is difficult to compute with abstract linear maps. I'll also comment that I don't really "get" abstract linear algebra (or anything abstract, for that matter). If I want to understand a liner map, I first sit down and remind myself how $2\times 2$ matrices work, then think about finite dimensional square matrices, then rectangular matrices, then linear operators in a separable Hilbert space. | |
| May 14, 2020 at 23:34 | comment | added | Nick C | One note: Considering that many/most students who take introductory linear algebra are engineers, these "secondary, technical devices" are going to be with us, whether we think they're pure-enough or not. | |
| May 14, 2020 at 22:22 | comment | added | user507 | This seems like an "am I right?" question. | |
| May 14, 2020 at 21:25 | history | asked | Kostya_I | CC BY-SA 4.0 |