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February 13th, 2011, 07:25 PM   #1
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Help with differential systems dealing with linearity and li

1) Verify the linearity principle for linear, nonautonomous systems of differential equations?
2) Consider all linear systems with two zero eigenvalues. which of these systems are conjugate? prove this.
3) Prove that a linear map T : R2 -> R2 is a homeomorphism if and only if it is nondegenerate?

I'm trying to figure these questions out for practice for an upcoming midterm. I don't know where to start or how to start any of them
I know that nondegenerate just means that the det =\= 0 and that the linearity principle is "if X' = AX is a planar linear system for which Y_1(t) and Y_2(t) are both solutions, then the function aY_1(t)+bY_2(t) is also a solution to this system."
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February 14th, 2011, 12:03 PM   #2
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Re: Help with differential systems dealing with linearity an

Hello lu5t.

I'd like to help you out a bit, but I am not quite sure of your definitions, so I will state any assumptions I make along the way. The first question on your list is a simple verification---it doesn't depend on the number of equations in the system, only on the properties of the operators. Now, if you have a linear system of differential equations, then what this says is that all the operators in your system are linear differential operators. So, to verify the principle of linearity, simply take a linear combination of (presumed) solutions to your system, plug it into the system, use the linearity of the operators, and conclude that the combination is a solution as well.

For the second question on your list, you will need to define the sort of conjugation you are referring to. But in any case, let's take the setting of linear algebra as an example. Two matrices A and B representing two linear transformations on a linear space are similar if they are conjugates by some non-degenerate matrix P, i.e. if such P exists so that B = PAP?¹. Now, you should prove that similar matrices have the same eigenvalues. Once you have done that (and it's not a hard calculation, just write it down), then you can ask about the number of conjugacy classes of matrices with two zero eigenvalues.

The third question is actually not that bad, if I am understanding it correctly. I am interpreting the word "non-degenerate" as meaning "non-singular", which is what your comments suggest. If you can prove that every linear operator on a finite-dimensional (topological linear) space is automatically continuous, then you are more or less done. Now, if your linear operator is non-degenerate, then this means it is a bijection, so the only thing left is to prove that the map is open (or that the inverse is continuous). But this is really almost trivial (and I hate to use that word because it sometimes sounds belittling) once you have proven the continuity of the operator. Conversely, if your linear operator is a homeomorphism, then by definition it is bijective, hence non-degenerate.

Let me know if anything I've written is unclear, and I wish you luck on your upcoming exam!

-Ormkärr-
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February 14th, 2011, 01:27 PM   #3
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Re: Help with differential systems dealing with linearity an

Thanks for the reply.

I actually ended up doing what you mentioned for number 1 and number 3.

As for the definition of the kind of conjugate that I was referring to, in my book it has: Given x'=ax and x'=bx have flows phi_a and phi_b. They are conjugate iff there exists a homeomorphism h: R^2 -> R^2 that satisfies phi_b(t, h(X_0))=h(phi_a(t,X_0))
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February 17th, 2011, 04:35 PM   #4
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Re: Help with differential systems dealing with linearity an

Hi again.

I forgot about your problem for a while ... did you end up figuring out the last bit about conjugation? Let me know how the proof turned out, or if you need any more help on it.

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