Is Taking Linear Algebra with DiffEq Better Than Taking It After?

[Null_]

By the beginning of the fall semester I will have completed Calc I-III (multivariable). I will be taking DiffEq in the fall.

I'm an engineering major aspiring to minor in math. Would taking LA with Diff EQ be better than taking LA after Diff EQ? My university does not require LA for engineering majors. If I do decide to take LA this fall, I'll have my heaviest load yet.

 

[Jack21222]

I don't think it matters too much the order in which you take DiffEq and LA. Of course, the sooner you take the math classes, the earlier you'll have the knowledge to apply to your engineering classes.

 

[Msh1]

Linear Algebra is very fundamental for studying more advanced mathematics. I am surprised how you were allowed to take Multivariable Calculus without having a course in Linear Algebra. Differential Equations on the other hand is not heavily used in more advanced math specially. So I would say take LA first then EQ or take both at the same time they are not really related after all.

 

[Dembadon]

At my university, LA is required for EE and ME and has Calc III as a prerequisite. As long as you have the prerequisite(s) covered, I don't see an issue. As far as workload goes, though, you know your limits better than we do. The best I have is an anecdote from a study partner; he is taking LA and DiffEq concurrently and hasn't complained about his workload. I can ask him if you'd like.

 

[CyberShot]

Linear Algebra is very fundamental for studying more advanced mathematics. I am surprised how you were allowed to take Multivariable Calculus without having a course in Linear Algebra. Differential Equations on the other hand is not heavily used in more advanced math specially. So I would say take LA first then EQ or take both at the same time they are not really related after all.

There is nothing in multivariable calculus that requires LA at the undergrad level. It's best to take LA before Diff Eq's, though because you'd need it when it comes to systems of DE's, and when you talk about resonance and the corresponding eigenvalue/vectors.

 

[mathwonk]

Anything can be taught without the proper background and in the old days multivariable usually was taught without LA. The point is you cannot understand multivariable calculus as well without understanding linearity. For that reason there was a revolution in the 1960's introducing LA before several variable calc.

E.G the point of differential calculus is that the derivative IS a linear map. And the correct statement of the chain rule is that the derivative of a composite of two maps is the composite of their derivatives.

They are useful as well in understanding the implicit function theorem which says that for a map of locally constant rank, the tangent space to the level set of f, is the level set of the derivative of f. I.e. the kernel of the derivative of f is tangent to the set {f=c} where f is constant.

Since linear maps can be calculated by multiplying matrices and the matrix of a composition is the matrix product of their matrices this insight can be omitted at the cost of making the subject computational instead of conceptual.

The answer to when should I study LA, is always "now". Once you learn it, it is useful for everything else.

 

[CyberShot]

Anything can be taught without the proper background and in the old days multivariable usually was taught without LA. The point is you cannot understand multivariable calculus as well without understanding linearity. For that reason there was a revolution in the 1960's introducing LA before several variable calc.

E.G the point of differential calculus is that the derivative IS a linear map. And the correct statement of the chain rule is that the derivative of a composite of two maps is the composite of their derivatives.

They are useful as well in understanding the implicit function theorem which says that for a map of locally constant rank, the tangent space to the level set of f, is the level set of the derivative of f. I.e. the kernel of the derivative of f is tangent to the set {f=c} where f is constant.

Since linear maps can be calculated by multiplying matrices and the matrix of a composition is the matrix product of their matrices this insight can be omitted at the cost of making the subject computational instead of conceptual.

The answer to when should I study LA, is always "now". Once you learn it, it is useful for everything else.

Yes, but you must agree that LA is just another way of looking at things. It's introduced because it's convenient and allows the mind to 'see' the connection between two concepts. It's a tool, but not a requirement.

 

[Null_]

Anything can be taught without the proper background and in the old days multivariable usually was taught without LA. The point is you cannot understand multivariable calculus as well without understanding linearity. For that reason there was a revolution in the 1960's introducing LA before several variable calc.

E.G the point of differential calculus is that the derivative IS a linear map. And the correct statement of the chain rule is that the derivative of a composite of two maps is the composite of their derivatives.

They are useful as well in understanding the implicit function theorem which says that for a map of locally constant rank, the tangent space to the level set of f, is the level set of the derivative of f. I.e. the kernel of the derivative of f is tangent to the set {f=c} where f is constant.

Since linear maps can be calculated by multiplying matrices and the matrix of a composition is the matrix product of their matrices this insight can be omitted at the cost of making the subject computational instead of conceptual.

The answer to when should I study LA, is always "now". Once you learn it, it is useful for everything else.

Interesting. I'm really surprised that my school doesn't require LA for engineering or science majors, and mostly only math majors take it. I.e. hardly anyone in our DiffEQ classes has taken LA.

Thanks for the responses. I've decided to take it in the fall (with DiffEQ) and see how it goes. There are three levels of linear algebra offered, and I've decided to take the highest one offered, which is proof based. I just ordered a book from amazon that was recommended in some previous threads about it which I'll browse over the summer.

 

[snipez90]

I took a theoretical ODE course this past quarter and even there I don't think I really needed linear algebra. OK so you need to know how to work with matrices at a 2x2 level: determinants, eigenvalues, and so forth. But these are the basics, and one could argue that basic LA concepts such as linear independence and eigenvalues can be motivated by the introduction of the Wronskian or systems of differential equations, respectively.

You do need LA to understand MV Calc well at a level beyond Stewart's text. But if you were planning to do analysis, then you probably already know that LA is a must.

 

[HallsofIvy]

Interesting. I'm really surprised that my school doesn't require LA for engineering or science majors, and mostly only math majors take it. I.e. hardly anyone in our DiffEQ classes has taken LA.

I'm sorry to hear that. The entire theory of "linear differential equations" is based on Linear Algebra- the set of all solutions to a linear homogeneous d. e. form a vector space so we only need to find a basis. And, of course, much of the concepts in linear d.e.s can best be expressed as matrix equations. I was responsible for making Linear Algebra a pre-requisite for Differential Equations at my college.

Thanks for the responses. I've decided to take it in the fall (with DiffEQ) and see how it goes. There are three levels of linear algebra offered, and I've decided to take the highest one offered, which is proof based. I just ordered a book from amazon that was recommended in some previous threads about it which I'll browse over the summer.

 

[Sankaku]

The answer to when should I study LA, is always "now".

Great line :-)

Once you learn it, it is useful for everything else.

I am always surprised when I hear that some Physics or Engineering programs don't require a separate course in Linear Algebra.

 

[mathwonk]

elementary differential equations (constant coefficient linear ordinary d.e.'s) is almost the same subject as linear algebra, and might be an even better introduction than a course on linear algebra. i.e. the basic linearity result in ode is the statement that for certain types of d.e.'s, the general solution of the inhomogeneous equation is the sum of a particular solution, and the general solution of the corresponding homogeneous equation.

In linear algebra that says the complete solution of an equation of form Tv = c, is all sums of form w+a, where a is a specific solution of Ta = c, and w is any solution of Tw = 0. I.e. if Ta = c, the the full solution set of Tx = c, is "a + kerT".The so called jordan form of a nilpotent linear operator over the complex numbers, is essentially (copies of) the matrix of the differentiation operator acting on polynomials of degree n.

However, with no tutorial on abstract linearity, it is harder to really understand why the solutions of equations like (D-r)^n (f) = 0, should have form x^k.e^(rx).

The point is that solutions of T^n y = 0, are the inverse image under T, of solutions of T^(n-1) y = 0. But without thinking in terms of operators, this does not leap to mind.

Again the difference without LA or with LA, is the difference between learning mindlessly a procedure, and understanding where it comes from. Knowing concepts of LA is not a requirement for taking many courses in later material, but it is a requirement for understanding them.

 

[mathwonk]

After teaching these subjects for many years, it is my opinion however that diff eq is the main application of linear algebra, and the two should be taught together. I.e. many LA courses which do not include applications to LA do a disservice in my opinion, just as courses in diff eq which do not use LA concepts do a disservice. As for advanced calculus, that is simply a hodge podge of complicated formulas, which are almost impossible to grasp, when LA is not used. Fortunately physicists and engineers do know the physical meaning of some advanced calculus constructs, such as Greens theorem and Gauss theorem, and sources and sinks of vector fields, and that probably saves them in those cases.

 

[Null_]

Mathwonk, thanks for your insight. I can't stand just being given a formula (which has sadly been the case a little too often). It seems that taking LA with diff eq will be good then. There are two higher level LA courses, one is a 3XX and the other is a 4XX. I was planning on taking the 4XX because it is proof based, which I honestly enjoy. The 3XX is application based.

I'm thinking that it will be easier to learn applications on my own than learn proofs. Does anyone have any words of caution regarding the selection?

 

[symbolipoint]

After teaching these subjects for many years, it is my opinion however that diff eq is the main application of linear algebra, and the two should be taught together. I.e. many LA courses which do not include applications to LA do a disservice in my opinion, just as courses in diff eq which do not use LA concepts do a disservice. As for advanced calculus, that is simply a hodge podge of complicated formulas, which are almost impossible to grasp, when LA is not used. Fortunately physicists and engineers do know the physical meaning of some advanced calculus constructs, such as Greens theorem and Gauss theorem, and sources and sinks of vector fields, and that probably saves them in those cases.

Putting both of these courses into one single combined course seems a bad idea. One or the other of those parts will suffer. I was a student in such a course. The differential equation portion was mostly good, but most of the linear algebra portion was too confusing be be understood. I did somewhat better studying some of (not much, just "some of") linear algebra recently on my own from a Hostetler/Edwards Linear Algebra book.

 

[WatermelonPig]

Hey I'm about to finish all three usual engineering maths. I'd recommend you take LA before DiffEq because the concepts of linearity are very important when understanding solution sets in DiffEq (homogeneous/nonhomogeneous). It is the same with Calc 3 but obviously you've already done that. And yeah LA usually isn't required for engineers but you should still take it its really useful. Or you could just skim Lay's book before starting.

 

[mathwonk]

try reading the first 20 pages or so of my notes for math 4050, #7 free notes on this web page: http://www.math.uga.edu/~roy/

I combine ode with LA there.

 

[AlephZero]

Knowing concepts of LA is not a requirement for taking many courses in later material, but it is a requirement for understanding them.

Amen!

(But understanding the concepts is not quite the same as being able to "talk the talk". It's years since I needed to write any math using the term "kernel", but I use the idea of what a kernel IS constantly.)

 

[Null_]

try reading the first 20 pages or so of my notes for math 4050, #7 free notes on this web page: http://www.math.uga.edu/~roy/

I combine ode with LA there.

Thanks! I've bookmarked it to go over this summer..I have to make sure I focus on this semester's material first as finals are not too far away!