Lesson 10: Vector Spaces
Preview
You have likely by now noticed that the real-world has many mathematical structures in it (e.g., the spirals found on some mollusk shells). But you may not know of just how ubiquitous these structures are. In this lesson we’ll discover that sound—and music—have mathematical structure, and in particular, a mathematical structure shared with geometry, differential equations, and linear algebra, among other connections: that of a vector space.
In Module I we’ll explore the arithmetic of matrices and arrive at 10 arithmetical properties satisfied by matrices.
In Module II we’ll specialize to nx1 matrices—vectors—and explore those 10 properties visually by thinking of vectors as arrows.
In Module III we’ll generalize those 10 properties and use them to define vector spaces.
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 10.pdfClass Notes A
Class Notes B
Class Notes C
Class Notes D
Class Notes E
Class Notes F
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 10 - Practice Problems.pdf