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Preview
Preview
In this lesson we’ll finally resolve the open question, How do we determine when Ax = b has either no solution or an infinite number of solutions without first solving the system to find out? The answer will involve some of the vector space concepts we’ve learned about in the previous two lessons, as well as investigating a new question: For what b does Ax = b have a solution?
In Module I we’ll prove that the set of b for which Ax = b has a solution has to do with the columns of the matrix A.
Finally, in Module II we’ll develop a technique to determine when Ax=b has either no or infinitely-many solutions by combining what we learn in Module I with some of the vector space concepts we’ve learned before.
Review
Review
Learn
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 Notes
Lesson 12.pdfClass Notes A
Class Notes A
Class Notes B
Class Notes B
Class Notes C
Class Notes C
Class Notes D
Class Notes D
Reflect
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
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 12 - Practice Problems.pdfPage updated
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