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)?
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.
