Practice problems for Exam 1 discussion area
Question from Katherine Mathews
I have a general question... When finding a real solution from a complex solution, do you only need to use one of the eigen-vectors? Each example that was done in class, you only completed the real solution from one eigen-vector and I am not sure if that was done for time sake or if that is the complete solution. If the complete solution can be determined from one eigen-vector complex eigen-vector, does it matter which one you pick?
Answer from Steve Bell:
I need to get two linearly independent solutions to the system from a conjugate pair of complex roots. I can squeeze TWO REAL SOLUTIONS from just one complex solution. When I find them, I forget where they come from and use them to form the general solution. If I were to use the other complex eigenvalue from a conjugate pair to get two real solutions, the real part would be the same as the one I got for the other root, and the imaginary part would be MINUS the one I got for the other root. So I would get the same general solution from the two real solutions I would get from the other complex root. Hence, no need to deal with it.
Question:
When solving a non-homogeneous solution, if we prefer using method of variation of parameters instead of method of undetermined coefficients, is it necessary to know how to use the latter in a question, or is it sufficient to know/understand one of the two methods (either one)?
Answer from Steve Bell:
The Method of Undetermined Coefficients is much easier than the Method of Variation of Parameters -- when all goes well, i.e., when the trial solution is obvious and no part of it solves the homogeneous system. However, when things do not go well, the Method of Undetermined Coefficients goes nowhere, but the Method of Variation of Parameters works fine, but is rather hairy. So all engineers need to know both.
On an exam, I could only ask rather easy questions about 2x2 systems to make them fit into a 50 minute exam. If you understand the 2x2 examples I did in class, you'd be ready.
Question from Kees
Is the row rank always equal to the column rank, e.g. is the rank of the matrix = row rank = column rank?
Answer from Steve Bell:
Yes, it is a fact that the dimension of the row space is equal to the dimension of the column space. See Theorem 6 on p. 286.
Question from Kees
Would it be a good idea to go over the different types of problem applications, e.g. fertilizer in tanks, electrical circuits, mass/spring damper system etc, or is that outside the scope of the exam?
