Homework 1

[goodwine]

Reading: Chapter 6, sections 6.1-6.4.

Exercises: 6.2 (only A_2, A_5 and A_9), 6.5 (A_5 and A_9 A_7 and A_8 only). I changed A_5 to A_7 and A_9 to A_8

Pendulum: This problem starts the theoretical work you need to do for the pendulum project. Determine the equation of motion for a pendulum of length l with a mass attached to the end with mass m as illustrated in the following figure where tau is an applied torque. Is the equation linear or nonlinear? If the angle is small so that you can assume sin(t) is approximately t, if you replace any sin(theta) by theta and any cos(theta) by 1, is the resulting equation linear or nonlinear? Using the same assumption (small angle) what is the equation if the angle is zero when the pendulum is hanging straight down instead of up?

"Determine the equation of motion" means find the differential equation. You do not have to solve it.

[goodwine]

Someone asked me:

Do we need to compute the eigenvalues and eigenvectors for each homework problem by hand or can we use Matlab?

You can do them however you want on homeworks, but you need to be able to do them by hand on exams.

[rboyle1]

Is the inverse of Theorem 6.1 true? i.e. Is it still possible for a 3x3 matrix to have 3 lin. ind. eigenvectors even if it does not have 3 real distinct eigenvalues?

[goodwine]

Is the inverse of Theorem 6.1 true? i.e. Is it still possible for a 3x3 matrix to have 3 lin. ind. eigenvectors even if it does not have 3 real distinct eigenvalues?

Yes, it is possible for a 3x3 matrix to have 3 lin ind eigenvectors even if it has some repeated roots. For that to happen you should see more than one row of zeros when you do the row reduction.

[goodwine]

Is the inverse of Theorem 6.1 true? i.e. Is it still possible for a 3x3 matrix to have 3 lin. ind. eigenvectors even if it does not have 3 real distinct eigenvalues?

Yes, it is possible for a 3x3 matrix to have 3 lin ind eigenvectors even if it has some repeated roots. For that to happen you should see more than one row of zeros when you do the row reduction.

I should add, though, that it's possible for a 3x3 matrix to have repeated eigenvalues and NOT have 3 lin ind eigenvectors. It might be either case. In this week's homework, it happens to be the case that you can find linearly independent ones.