Abstract linear algebra versus applied

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Discussion Overview

The discussion centers on the comparison between abstract linear algebra and applied linear algebra, particularly in the context of preparing for graduate studies in mathematics. Participants share their experiences with different courses and express concerns about the timing and importance of taking proof-based linear algebra before advancing to more complex topics like differential geometry and differentiable manifolds.

Discussion Character

  • Exploratory
  • Debate/contested
  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant reflects on their experience with applied linear algebra, noting a lack of learning due to not attending class, and questions the necessity of taking abstract linear algebra before graduate school.
  • Another participant suggests that having taken an applied course, the abstract course may be easy, especially since many students in the abstract course have not encountered matrices before.
  • Concerns are raised about the importance of being proficient in linear algebra, as it is frequently used in advanced courses, including smooth manifolds.
  • Participants discuss the timing of taking the abstract linear algebra course, with some recommending it be taken early, while others suggest it may not be critical since the individual is in their last year of undergraduate studies.
  • One participant shares their experience of taking proof-based linear algebra early and feeling unprepared later, indicating the importance of timing in relation to other courses.
  • Another participant expresses enthusiasm for complex analysis, describing it as an engaging subject that differs significantly from typical analysis.

Areas of Agreement / Disagreement

There is no clear consensus on whether to take abstract linear algebra before graduate studies, with some participants advocating for its early completion while others suggest it may be less critical given the individual's background and course schedule. The discussion reflects a mix of personal experiences and varying opinions on the importance of timing and proficiency in linear algebra.

Contextual Notes

Participants express differing views on the necessity and timing of taking abstract linear algebra, with some emphasizing its foundational role in advanced mathematics, while others consider the individual's prior experience and course load. There are also references to specific courses and teaching styles that may influence learning outcomes.

jmnance
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I have taken applied linear algebra http://courses.illinois.edu/cis/2009/fall/schedule/MATH/415.html?skinId=2169"

and didn't learn anything really since i never went to class (yeah yeah yeah I know). I am taking intro to abstract algebra 1 and 2 this year. My friend took the abstract version of linear algebra http://courses.illinois.edu/cis/2009/fall/schedule/MATH/416.html?skinId=2169"
and said he learned linear algebra really well because of how abstract it was. I plan to go on to grad school for math and am wondering if I should take this abstract linear algebra course before I attend or if I will have learned the meat and potatoes by means of taking the under grad abstract algebra courses and a grad level abstract algebra.
 
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Wow.. I took honors linear algebra (Prof.Ando) last semester
I think if you've taken 415 then you don't need take this course
it's a proof-based course though
but about 1/2 students taking 416 have not even seen matrix before...
so the course will probably very easy for you since you already know Linear algebra stuff
 
Was honors lin alg 416? I know that Ando taught that. Was Rick Barber in your class?
 
yeah, it's honors 416. I know Rick, but he probably don't know me.. lol
Still I think the course will be very easy for you.
Especially next fall 416 will be a requirement for all incoming math major freshmen,
so I think the course will be even easier.
We use Paul Halmos' Finite Dimensional Vector Space last semester,
and you probably want to check it out on Google book..
 
I would definitely recommend taking the proof based linear algebra class if you have time. It is very important to be proficient at linear algebra, as it always pops up, and if you aren't comfortable with it, it will give you trouble on top of the material you're trying to apply it to. My smooth manifolds course (calculus on manifolds) used linear algebra heavily, and it really hurt most of us because we weren't comfortable doing abstract things with determinants, matrices, etc.
 
Jin8- That is funny. Rick is my best friend!
n!kofeyn-so where should I fit it into my schedule? I mean what types of courses should it precede? I plan on taking a differential geometry course and a differentiable manifold's course in grad school. I guess I should take it before then. Would it hurt to take it my very last semester?
 
You should take it as early as possible, since linear algebra really is basic mathematics that pops up everywhere, even 'early' courses like multivariable calculus/analysis use it heavily.
 
jmnance said:
n!kofeyn-so where should I fit it into my schedule? I mean what types of courses should it precede? I plan on taking a differential geometry course and a differentiable manifold's course in grad school. I guess I should take it before then. Would it hurt to take it my very last semester?

The advice to take it as early as possible doesn't really apply to you since it is your last year (if I remember correctly from your other post) and the fact that you've already had an applied linear algebra course. Since you'll be taking analysis this fall and topology next spring, it won't be a problem if you take it during the spring if you can't fit it in during the fall. You definitely will benefit taking it before you graduate and before you take differential geometry and differentiable manifolds. The differentiable manifolds is the same course I mentioned above (smooth manifolds or calculus on manifolds) and relies on a lot of linear algebra. Being comfortable with linear maps, rank of matrices, determinants, general matrix stuff, etc. will help you tremendously in that course as well as the differential geometry course.

I actually took my undergraduate school's proof-based linear algebra course early on (two years before I graduated), and this ended up hurting me a little as I didn't fully appreciate the theory and had forgotten a lot of it by the time I got to courses that actively utilized it.
 
again, another great comment. Thank you all who have commented. Looks like I'm going to take it my last semester as an undergrad (along with grad abstract algebra and undergrad complex analysis ugh..)
 
  • #10
jmnance said:
again, another great comment. Thank you all who have commented. Looks like I'm going to take it my last semester as an undergrad (along with grad abstract algebra and undergrad complex analysis ugh..)

Complex analysis is very cool. I think of it as some type of exotic math, because it really works so different than typical analysis. It is very fun, and the proofs are so much better. Say goodbye to boring and ugly epsilon-delta proofs.
 
  • #11
oh man you're preaching to the choir! I bought a copy of Visual Complex Analysis and can't put it down... sooooo cool!
 

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