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MATH 122.C (Spring 2022)

Applied Linear Algebra

Syllabus for Math 122 (3 pages)

Homework

Video: How I expect you to do homework problems
If your textbook has not yet arrived, please refer to the scanned pages of the Homework assignments of the first four covered sections, as well as some required pages from Sec. 1.5:

Please note that no more pages (including answers to problems from Sec. 1.5) will be posted. It is your responsibility to order the textbook in such a way that it arrives during the first two weeks of the semester.

Note about the prerequisite material from Calculus II

The material that you must know from Calculus II is:
1. Definition of vectors in 2D and 3D;
2. Geometric addition and scalar multiplication of vectors (this includes parallel vectors);
3. Equations of lines in 2D and of lines and planes in 3D.

If you do not remember some of this material, you must review it on your own.
- Topics 1 and 2 are reviewed in Sec. 12.2 of your Calculus textbook by Stewart,
and also in Secs. 2.1 and 2.2 of your Linear Algebra textbook. (You should focus
only on these topics in Secs. 2.1 and 2.2 and disregard other topics there.)
- Topic 3 will be very briefly reviewed in Sec. 3.1.
It is also reviewed in Sec. 12.5 of Stewart's Calculus and in Sec. 2.4 of your Linear Algebra book.

Instructions for doing an extra credit assignment (please read them before you attempt to do one)

What to do if you received a low grade on a quiz (and want to improve your performance)

My policy on Projects (please read it as soon as the first Project is assigned)

Projects must be turned in in class on the due date. For Projects 2, 4, and 5, you must also email me Matlab codes, as described in the instructions for these Projects.
If you have to submit your project later than the deadline, you must contact me no later than on the due day (before our class) and ask for an extension. Cases involving extraordinary circumstances (where you can demostrate a substantial component beyond your control) will be handled individually. Other cases will be handled as explained below.
Late submissions on the same day (with approved extension request) (i.e., until midnight) will be graded out of 80%.
Late submissions during the following two days (with approved extension request) will be graded out of 50%.
Any later submissions, or submissions without a prior request for an extension, will not be graded.
Statistics of academic integrity violations on Tests, Quizzes and Projects

Project 1 Assigned on: 02/02 Due: Wednesday, 02/16, before class

Note: Project 2 will be assigned on Project 1's due day

Project 2 Assigned on: 02/16 Due: Friday, 03/04, before class

Skeleton of the code that I encourage you to use (or at least consult) when writing your own code for step 4
Help session: Please email me before 9 pm on Monday, 2/21, and request a help session. If I get more than 3 requests, I will hold a help session. Otherwise, I will invite you to come for individual help.
If you have not used Matlab before, you need to install it first on your computer. Go to the installation instructions link below and install Matlab before you come to Help session. The installation process takes about 30 minutes, and so you will not be able to complete the installation at the Help session and obtain meaningful help from the instructor.
If you are new to Matlab, please study the first two of Matlab resources listed below.
If you want to work with a partner on this Project but don't know anyone in class who can be your partner, send me an email. If I have more than one such a request, I will introduce partner-seeking people to one another. You are not obligated to follow my suggestion, and you are not obligated to continue working with the same partner on all projects. This is just an opportunity that you may want to explore.
Do not forget to email me the Matlab code, as well as follow all other directions on page 1 of the Answer Sheet.

Project 3 Assigned on: 03/14 Due: Monday, 03/28, before class

Example for Appendix
Project 3 is due around the time of Test 2. Therefore, do not wait until the last minute to start working on it. If you decide to wait, you will end up having to both prepare for the test and work on the project, which will be detrimental to both the test and the project grades.

Project 4 Assigned on: 03/28 Due: Wednesday, 04/13, before class

You must also email me the Matlab code, as well as follow all other directions on page 1 of the Answer Sheet.
code for Exercise 1
codes for the Example after Exercise 1: version 1, version 2
codes for Bonus-2: example 1, example 2
Solutions: code(s) for Parts 1 -- 3 , code for Bonus-2

Project 5 Assigned on: 04/13 Due: Monday, 05/02, before class

Project 5 is due around the time of Test 3. Therefore, do not wait until the last minute to start working on it. If you decide to wait, you will end up having to both prepare for the test and work on the project, which will be detrimental to both the test and the project grades.

Solutions to Quiz 4
Preparation sheet for Test 1

Test 1 will be given after we cover Sec. 3.1
Test 1 will be held from 6:30 pm to 7:45 pm on Monday, March 17, in Innovation E342.
(The two people who take the exam in the alternative format with take it on Friday, 3/18, from 3:30-5:30 in Innovation E425 (a conference room).)

Solutions to Test 1
Solutions to Quiz 7
Preparation sheet for Test 2

Test 2 will be given after we cover Sec. 3.7

Example of how NOT to prepare a formula sheet
Test 2 will be held from 6:45 pm to 8:00 pm on Monday, April 11, in Innovation E432.
The two people who take the exam in the alternative format with take it at the same time in Innovation E325 (a conference room on the 3rd floor).

Solutions to Test 2
Preparation sheet for Test 3

Test 3 will be given after we cover Sec. 4.5
Test 3 will be held on Fiday, April 29 from 5:45 and 7:00 pm in Innovation E430.
People who will take the test in the "alternative format" (as before), will do so on the same day from 5 to 7 pm in E432.

Solutions to Test 3
Solutions to Quiz 10

Quiz 10 (the last one) will be given either on Monday or Wednesday of the last week of classes (i.e., May 2 or 4). This quiz will be based on Part I of Sec. 4.7.

Preparation sheet for the Final Test

Important suggestion: Begin the preparation from the end of the list (i.e., Sec. 4.7) and work through it in reverse order. I recommend this because problems from later sections (and especially that or those from Sec. 4.7) will be assigned more weight than problems from early sections.

Various types of proofs (most of which you may want to avoid :-) )

Lecture notes (On a computer, each of these files should open in a new window or tab. On a mobile device, you need to open the file in a new tab to view it; otherwise you can see only part of its first page.)

Sec. 1.1

With comments

Sec. 1.2

Example 7 of Sec. 1.1 rewritten using the concept of Echelon and Reduced Echelon forms

Websites showing steps of reduction of a matrix to REF. They have other matrix calculators, too, some of which may be useful later in this course.

Linear Algebra Toolkit (link courtesy of Rex Calabrese, who took this course in Spring 2014)
Symbolab; this program even allows symbolic manipulations, much like Mathematica (link courtesy of Olivia Healy, who took this course in Spring 2018)

Note: Examples in the textbook show an INCORRECT way of going from the EF to REF. Namely, they allow one to use, say, row 2 before row 3 to eliminate coefficients in row 1. Please do NOT follow this incorrect approach. Instead, follow the approach shown in class and in the above two websites.

Sec. 1.3
Sec. 1.5

Clarification on MUST-READ textbook Examples for Sec. 1.5

One of these examples clarifies the must-know concept of "MATRIX FORM" of a linear system.
Note that the concepts of "vector form" and "matrix form" are COMPLETELY DIFFERENT.

With comments

Matlab demo codes illustrating matrix addition and scalar multiplication

Sec. 1.6
Before working on Sec. 1.7, make sure to review Topics 1 and 2 of the prerequisite material, as emphasized at the top of this page as well as at several places of the Homework page.
Sec. 1.7 - Part A (Linear dependence and independence of vectors)

Matlab file linear_combination_of_vectors.m , illustrating that as long as 2-dimensional vectors v1 and v2 are linearly independent, any third vector in the same plane is their linear combination. You do NOT need to be familiar with Matlab to be able to use this file. You just need to save it on your computer and then open it (or run) in Matlab. When you run this file, it will prompt you to enter any three vectors v1, v2, v3 (where v1 and v2 are linearly independent) and will then produce a plot illustrating that v3 = c1*v1 + c2*v2. It will also make comments along the way.
The website quoted under the link for Sec. 1.2 has a detailed and friendly calculator of linear (in)dependence of vectors. Enjoy! :-)

Sec. 1.7 - Part B (Singular and nonsingular matrices; and More proofs)
Sec. 1.8
Sec. 1.9
Before working on Sec. 3.1, make sure to review Topic 3 of the prerequisite material. (It is assumed that you have already reviewed Topics 1 and 2 before working on Sec. 1.7.)

Sec. 3.1

Sec. 3.2

Sec. 3.3

Additional example illustrating the concept of Range

Sec. 3.4

Sec. 3.5

Sec. 3.6

Example of Gram-Schmidt orthogonalization as per above Notes (different from textbook)

Sec. 3.7

Sec. 3.8

Sec. 4.1

Sec. 4.2

Sec. 4.4

Sec. 4.5

Sec. 4.7 - part I

Sec. 4.7 - part II

Application: Oscillations of the H₂O molecule

The concept of Singular Value Decomposition (SVD)

Instructions how to install Matlab on a student-owned computer at UVM. (Just in case: This link is supposed to open in a new tab or window. If for any reason it doesn't (in which case it wouldn't open at all), manually instruct it to open in a new window or tab.)

As you are going through the installation, stay tuned to see and note down the License number (or an Activation key, if for some reason the installer presents it instead of the License number). See the next bullet for its usage.

Instead of installing Matlab on your computer, you may run your programs on Matlab's server online via this link: https://matlab.mathworks.com/. You will need your Mathworks userID and password, which you created when signing up with Mathworks during Matlab installation process. This is where you will need to enter the License number (or an Activation key).

MATLAB resources

A link to Matlab Onramp - a hands-on getting-started tutorial. You will need to login with your Mathworks userID and password to access it.
MATLAB Primer by K. Sigmon

This manual is 30 years old, yet it is one of the best written getting-started manuals that I know of. You need to read only up to the end of p. 9 plus section 18 (graphics).

MATLAB tutorials by E. Neumann:

Getting started with MATLAB

Programming in MATLAB

Using MATLAB in Linear Algebra

Numerical Linear Algebra

Numerical Analysis with MATLAB

MATLAB Primer by S. Nakamura. This is Chapter 1 from the book "Numerical Analysis and Graphic Visualization with MATLAB," 2nd ed.

Links of interest:

History of matrices and determinants. Also, the directory found at this site contains many interesting links about history of mathematics and related disciplines.

(Some) Things one can do with Linear Algebra