Linear Algebra and it's importance

[rubrix]

I took Linear Algebra a while ago, i passed but didn't do good. It's been a while i have come to realize how important it is and why i should not have blown up that course. Regardless, what is done is done, nothing can be done about that. However, i indent to "fix" by revising the course. To do so, i would like to know what *exactly* (please provide as much detail as you can) from Linear Algebra i need in following fields:

Physics

I know Linear Algebra is of great importance in Physics. Infact, most Physics department have it as prerequities. However, mine does not - Linear Algebra is not a requirement for my Physics department. They instead offer a Mathematics for Physicists course which goes in detail of Mathematical ideas required in Physics...a great deal of course is Linear Algebra from what i hear. I'll be taking this course next semester, so it will be short of a review.

Applied Mathematics

Most likely i will be going with Applied Mathematics so i would like to know how important is Linear Algebra in AM? I did Introductory DE course (which mostly had ODE) and it showed little to no correlation with Linear Algebra (although I'm sure there is some in higher ground). And i'll be doing Applied Differential Equations soon...and PDE at some point. Again *specific details* would be nice.

Abstract Algebra

Here, all Mathematics major are required to do two undergrad course on Abstract Algebra. Linear Algebra is not a prerequisites for Abstract Algebra yet in a quick overview i can see it has stuffs like matrices and homomorphism. Is that all that is from Linear Algebra or is there more? I would like to know in *very much detail*...that way i know if i should start reviewing asap or wait till i do Mathematics for Physicists course.

Thanks in advance.

P.S. if you respond to the thread, please check back, most likely i will have more questions for you ;)

 

[Gear300]

I'm not exactly sure what level of physics you're taking, but the use of linear algebra in physics is primarily due to vector analysis...this sort of analysis generally comes after one has covered the classical phenomenon with some calculus (though it can be used from the start). Its use becomes noticeable when you enter classical dynamics.

Abstract/Modern Algebra does not require linear algebra. This course is a more "fundamental" analysis of mathematics and its structures compared to linear algebra; in fact, modern algebra is more of a prerequisite to linear algebra (or for that matter, several other mathematical studies) than linear algebra is to modern algebra.

 

[comp_math]

Linear algebra provides you understanding of properties of physical systems, for e.g., finding the eigenvalues or "natural frequencies" of systems. Linear algebra also provides the background theory for development of solution methods to problems governed by PDEs, for e.g. finite differences, finite elements, etc. I can't comment on abstract algebra other than the fact it isn't linear algebra. There are many more.

 

[rubrix]

Abstract/Modern Algebra does not require linear algebra. This course is a more "fundamental" analysis of mathematics and its structures compared to linear algebra; in fact, modern algebra is more of a prerequisite to linear algebra (or for that matter, several other mathematical studies) than linear algebra is to modern algebra.

one of the reason i was asking for this (apart from the fact that i see some topics from Linear Algebra in Abstract Algebra book) is that my professors webpage says Linear Algebra is a prerequisites for Abstract Algebra (the official university webpage does not say so though). Furthermore, we have used matrices and basis in this abstract algebra course. Anyhow, thnx for the comment =)

Linear algebra also provides the background theory for development of solution methods to problems governed by PDEs, for e.g. finite differences, finite elements, etc.

can you be *specific* on what "background theory" you are talking about.

 

[comp_math]

can you be *specific* on what "background theory" are you talking about?

In linear algebra, one learns vector spaces, matrix norms, positive-definiteness, singular value decomposition, etc. Then, one uses these to study properties of numerical methods, for e.g., well-posedness and stability, or to study methods of solutions (direct, iterative, multigrid, etc.) for linear systems obtained from discretization of PDEs.

 

[Pyrrhus]

Linear algebra is extremely important. I see it everyday in my work, and courses (Econometrics, Operation Research, etc..)

 

[rubrix]

Linear algebra is extremely important. I see it everyday in my work, and courses (Econometrics, Operation Research, etc..)

That does not help though. As stated, i already know it is important...and you are just repeating it. You say "I see it everyday in my work" but i have no idea why/where/how/when/what your work is and how to correlates to what we do in class. Furthermore, you name two courses, which i guess you assume I'm familiar with (which I'm not)...yet you don't draw correlation between them.

What i am looking for is *specific* details to how what you do or what course you have studied (in particularly in higher Physics course, Applied Mathematics, and Abstract Algebra) and how it correlates to what we study in Linear Algebra. See comp_math's post above, that's what I'm looking for. See i can just google the specific topic/corelation he drew and learn more about it..but on topics you provided, well it's too vague.

 

[G01]

Linear Algebra is indeed important for vector analysis and matrices.

However, a physicist needs a good grounding in the theory of vector spaces as well. An understanding of vector spaces is extremely important if you want to study, for instance, quantum mechanics at an advanced level.

I mention this because I knew several physics majors who did well in linear algebra up to the point of vector spaces, but then struggled with the course. They shrugged it off, saying that the material covered in the study of vector spaces is abstract math and not applicable to physics.

Don't fall into this same trap. If you happened to struggle with vector spaces, focus on understanding the material, as it is very important for physics.

 

[snipez90]

The text I used to "learn" linear algebra started off with vector spaces, which I found pretty dull. This might have been one of the reasons why I didn't pursue Axler in depth, doing well enough just to earn an A (our calculus sequence ends with a 5-week excursion into linear algebra). I will have to relearn this at some point...

But I do actually have something to contribute to this thread, and this is to elaborate on why linear algebra might be a prerequisite for abstract algebra. An assistant professor who works in group theory at my university told us in one of the REU lectures that a lot of the mathematical objects in abstract algebra "try to be vector spaces", but fail to have properties as nice (ok so I could be totally incorrect, but I think this was the gist), so having a good understanding of vector spaces could be useful in learning abstract algebra.

 

[rubrix]

@G01, thnx, that helps :)

@snipez90, linear algebra - abstract algebra correlation you drew makes sense.

 

[Monocles]

Your professor is definitely correct about objects in abstract algebra "trying to be vector spaces." This is actually the springboard that myself and many other physics students I know used to jump into the ideas of algebra.