Lesson 13: Eigenvalues and Eigenvectors
Preview
We have seen many, many applications of linear algebra and matrices over the previous few lessons. In this last lesson in the applied linear algebra portion of the course we’ll develop two new concepts—eigenvalues and eigenvectors—that will expand dramatically the real-world phenomena we can apply linear algebra to. We’ll see this in action in the context of population renewal and demography in this lesson, and in the context of evolutionary genetics in the capstone that follows this lesson. Additionally, as we will see after the capstone, the eigenvalue-eigenvector framework will underpin the new foundations and applications of ODE theory we will develop in the final few lessons of the course.
In Module I we’ll motivate the eigenvalue-eigenvector framework by considering the problem of modeling population renewal. Then, we’ll define what eigenvalues and eigenvectors are and learn how to calculate them.
Finally, in Module II we’ll discuss a useful theorem connecting the vector space concepts we’ve learned to the eigenvalue-eigenvector framework.
Review
Learn
Work through the lesson notes below, consulting the videos below it when you get to the "See Class Notes" boxes. For your records, the annotated lesson notes are below the videos. Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson 13.pdfClass Notes A
Class Notes B
Class Notes C
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 13 - Practice Problems.pdf