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Preview
Preview
In the previous lesson we talked about how to solve Ax = b using elementary row operations. We discovered that the possible solutions fall into one of three categories: no solution, a unique solution, and infinitely many solutions. But using elementary row operations to solve linear systems takes a while, and we don’t know what solution category we’re in until after we’ve completed all the row operations. In this lesson we’ll introduce a faster and more algorithmic method—Cramer's Rule—for solving Ax = b. The method only works, however, if the system has a unique solution. Fortunately, the method gives us a quick way of checking beforehand if that’s true. That check (and the Rule) involves determinants, so we’ll start there first.
In Module I we’ll solve systems of two linear equations with two unknowns and discover that a specific quantity—what will turn out to be the determinant of the 2x2 coefficient matrix of a system—helps us determine whether a unique solution exists to such systems.
In Module II we’ll discuss the general definition of a determinant and also learn how to calculate determinants.
In Module III we’ll discuss Cramer’s Rule and its many ramifications.
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 9.pdfClass Notes A
Class Notes A
Class Notes B
Class Notes B
Class Notes C
Class Notes C
Class Notes D
Class Notes D
Class Notes E
Class Notes E
Class Notes F
Class Notes F
Class Notes G
Class Notes G
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 9 - Practice Problems.pdfPage updated
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