Differential eqns or linear algebra?

[renob]

Im about to finish up calc 2. which of these would be best to take assuming i can't take multivariable calc next semester?

 

[Newtime]

Im about to finish up calc 2. which of these would be best to take assuming i can't take multivariable calc next semester?

This depends on what you mean by "best." Are you a math major? an engineering major? Do you plan to go to grad school? What are you more interested in? etc.

 

[jeffasinger]

In general, Linear Algebra is probably more directly applicable overall, but as Newtime said, without more information about what you're planning on taking in the future, it's impossible to give a good answer.

 

[Jack21222]

If you're a physics student, then definitely DiffEQ. I haven't taken either class yet, and I'm finding that my lack of DE experience is killing me where my lack of linear algebra experience hasn't been a factor.

I'm a junior taking Classical Mechanics (lagrange's equations, etc), and that class is almost all DEs.

 

[renob]

I'm in engineering. I need to take both of them eventually, just wondering which one I'd be better prepared for since I've only gone up to calc 2

 

[scotto3394]

From what I've heard and experienced, you don't need calculus to learn Linear algebra and you need to be fairly good with calculus for Diff Eqs. So from what you've said I'm guessing either would be fine. However, you might want to learn partial differentiation before going to diff eqs. But, that's just my opinion.

 

[Ygggdrasil]

If you need to take both it makes the most sense to take linear algebra before taking differential equations. In fact, in some colleges linear algebra is a prerequisite for differential equations. This is because many topics in differential equations require concepts from linear algebra such as finding the eigenvalues of a matrix.

 

[renob]

alright thanks, good to know. I'll take linear algebra, then multivariable calc, then differential eqns.

 

[mathwonk]

i think linear algebra is more basic and more useful, and indeed almost essential in order to understand diff eq. on the other hand some people might find it useful to learn diff eq as an example of linear algebra at work, and then learn the linear algebra afterwards.

It is my theory that linear algebra was invented to systematize the techniques of elementary differential equations, i.e. linear ones. In fact, as pointed out in the notes for math 4050 on my website, the so called jordan form theory for matrices essentially says that all linear maps look like the basic linear differential operator D acting on a solution of an appropriate linear constant coefficient diff eq.

More precisely, jordan form says that matrices can be decomposed into pieces that correspond to polynomial factors of form (t-c)^r. These in turn correspond to solutions spaces of the differential operator (D-c)^r.

"Eigenvalues" arise in studying the easiest equations (D-c) which have solutions like e^(ct).

so really elementary diff eq is just linear algebra at work, and linear algebra is just linear diff eq made abstract.

Yet again, the most important linear map is D, differentiation, and the most important function is e^t, and the reason for this is that e^ct are the eigenfunctions of D.

So the two subjects are the same. Hence studying linear algebra without seeing the connection with diff eq is kind of dumb, and studying diff eq without having the linear nature of the subject pointed out is equally myopic.in the end, the fact that the general solution of a linear diff eq is of form f0 + g, where f0 is one particular solution and g is the general solution of the homogeneous problem is the main idea of both subjects. Some will learn this idea better in practice, i.e. from diff eq, and others may appreciate it abstractly, i.e. in linear algebra.

so everyone should know both subjects, even if only interested in one of them.

conclusion: take linear algebra but be sure the connection with linear diff eq is pointed out. consult for instance friedberg insel and spence, or at a higher level, pages 234-237 of chi han sah's abstract algebra (or my 4050 notes).