Linear Algebra

15-Math-2076

Department of

Mathematical

Sciences

This page is a work in progress! All information is subject to change (Last revised 5 January 2021)

Instructor Prof David A Herron
4514 French Hall, 556-4075 		My Office Hours
By appointment; email me. 		E-mail me at David's e-address
My web page is at David's w-address

Basic Course Information

Textbook Linear Algebra and its applications (5th edition) by David C. Lay (ISBN-13: 978-0321982384). Please note that you should get a version with the MyMathLab access code, as we will use Pearson's online homework system. Two options are:

  1. You can buy a physical textbook (bundled with the access code) from the University of Cincinnati Bookstore. (Other stores may have the same text, but please make sure you get a version with an access code.)
  2. You can purchase an eText + Access Code directly from the publisher, Pearson. (This may be the least costly plan!)
General Syllabus—Chapters 1 thru 7.

Online Course
This course is fully online. In place of traditional in person classes, we will have video content, online homework, online quizzes, forums for collaboration, and other online activities that—provided you and I both do our jobs well—will create an engaging learning environment. Please note that all exams will be on campus; see below for dates. Success in mathematics is like success in most things: it requires regular focused engagement with the subject. I will give you direction and support for your intellectual journey into linear algebra, but ultimately you must walk the path.

Links to Items Below

Important Dates

Course Structure

While this is an online course, I recommend that you think of it as a traditional MWF class. Each week we will cover approximately three sections in the text as described in the Weekly Syllabus. For each section, you should review the text material, watch the associated videos, and work the suggested exercises (which I call practice problems); again, see the Weekly Syllabus. Also, each week there will be both assigned/collected/graded online homework (usually due on, or before Thursday of that week) that covers the material (from the previous Friday and current Monday, Wednesday) along with a quiz (due on Friday) that covers the same material. See Course Grades for more information about the Quizzes, HomeWork, and Videos.

An online course requires a different approach from a traditional in-class course. While you spend no time commuting to UC and no time in a classroom, the lack of one-on-one interaction with the instructor requires extra effort studying the course videos and textbook. In order to fully understand the concepts being presented, you must actively watch the videos, and you may need/want to repeatedly reread the text and/or rewatch the videos.

Please note that just working the online homework will not fully prepare you for the course exams. These online exercises emphasize finding the correct answer which is one part of the course but is not the really important aspect of the course. I strongly encourage you to also work the Suggested Exercises. Please, work the Suggested Exercises.

Here is a detailed description of how to succeed in this course. Your primary focus should be on understanding the ideas.
  1. Scan the appropriate section of the textbook. For this reading, don't worry about understanding everything; you just want a quick look at the main topics covered.
  2. Watch the assigned videos. Here you do want to understand most everything that is discussed. Be active when you watch: take notes (you can use the associated pdf files that I provide), try to correctly answer all questions posed, be sure to understand both the ideas and the calculations (check my work!), pause when necessary. Have your book at hand, along with paper and pencil. Alternatively, you can download a pdf that has the video slides, and take notes directly on it.
  3. Now go back and carefully read the textbook paying extra attention to all topics that appeared in the video. Also, work through all the examples; here instead of following the book, work the example yourself and if you get stuck, then look at what is in the book. By now you should understand most everything.
  4. Watch the video again; this time it should make a lot more sense. Be certain that you understand all of it! If you answered any of the questions incorrectly, correct your misunderstanding. If you don't understand something on the video, you will probably be unable to work some exam question and for sure later material will be more difficult for you.
  5. Work the practice problems; see the Suggested Exercises at Weekly Syllabus. Here your primary goal is to understand what you are doing; once you understand, you will know how to get the right answer. Getting correct answers is important, but understanding the ideas is crucial, especially for material later in the course.
  6. Work the online HW. By now this should be really easy; really!
If you skip, or don't complete, one of the above steps, then later steps will be more difficult and/or more time consuming. You may have to repeat a step before moving on to the next step. If you cannot work the examples in the text, then reread the appropriate pages. If you cannot understand part of the video, then rewatch earlier parts and/or read appropriate parts of the text (or search the web for info).

If you don't understand the material or cannot work the suggested exercises, seek help: Ask your classmates, try the MASS Center, come to my office; just get help somehow.

Course Goals

The main course objective is to learn about linearity and especially linear transformations. A second objective is to understand some geometry in Euclidean $n$-dimensional space $\mathbb{R}^n$. Here is a brief list of some of the topics we will cover: systems of linear equations, matrices, Euclidean $n$-space and its subspaces, bases, dimension, coordinates, linear transformations, orthogonality, determinants, eigenvalues and eigenvectors, diagonalization.

Calculators: You may want a calculator for this class, but any inexpensive one will do. Unfortunately, the calculator on your cell phone will not work, since you will not be allowed to have it out during exams. Nowadays many calculators will do virtually all of the computations that we learn, but please do not rely too heavily on your calculator!

Canvas: Keep an eye on canvas. It will be used to post announcements, quizzes, some exams, answer keys, scores, and more.

Course Tools

Here are the major platforms that you need to become familiar with:

Your Course Grade

Your final grade will be based on three hour exams, a final exam, quizzes, homework and EdPuzzle participation. Here is the precise breakdown: Your grade will be determined solely from the data described above—there will not be any possible "extra credit".

Barring unforeseen extraordinary circumstances, there will be no makeup exams nor makeup quizzes; if you cannot take an exam, you should not expect to be able to make it up except in unusual conditions. If you have a valid reason for missing an exam, please speak with me about it before the exam and I will try to make arrangements. (Legitimate, documented exceptions—such as illness, with a doctor's note—need to be approved by me and will be dealt with by shifting the weighting to the other exams.) The exam dates are listed here.

If your Final Exam score exceeds one of your hour exam scores, then it will replace that exam score; so in this case, your Final Exam score will count for 38% of your final grade.

Quizzes
Weekly quizzes will be electronically assigned and collected via canvas; this should be your own work! It is your responsibility to take the quizzes on or before the due dates/times. Late quizzes will not be accepted. I will drop your lowest quiz score. There will be no make-up quizzes. There is a "warm-up" Quiz 0 that is available the entire first week of the class; please take it as soon as possible.

These quizzes will be "paper and pencil" work. You will download and print the quiz (from canvas), work the given problems, and then upload your work (thru canvas). Please note that these quizzes are timed: once you start the quiz, you will have a limited amount of time to upload your final answers; you can logout of canvas while taking the quiz, but the clock keeps ticking. There is plenty of time allocated for you to print the quiz, make a pdf copy of your work, and upload it; but use your time wisely. Late quizzes will not be accepted.

It is your responsibility to make sure that you allow time to create and upload a legible pdf of your work; failure to do so will mean that your work will not get graded meaning that you will get a 0 for that quiz. Please allow time for technical difficulties.

There are instructions posted in the Weekly Quizzes folder in blackboard that explain the quiz procedure. There is also an initial Quiz 0 that requires no work but allows you to learn the quiz download/upload procedure. Yes, the Quiz 0 points will count towards your grade.

Homework
Each section has plenty of practice problems (see the Suggested Exercises listed at Weekly Syllabus) that I strongly urge you to work through; these will not be collected nor graded. I encourage you to work with other members of the class on these problems. In addition, there will be homework assigned via MyMathLab (see Course Tools) that will be electronically collected and graded; this should be your own work! It is your responsibility to turn in the homework assignments on or before the due dates. Late homework will not be accepted.

Please work the Suggested Exercise along with the online HW. I believe that the online HW itself will not adequately prepare you for the in class exams. Each section will have some assigned online exercises, and these will be due each week on Thursday (so, the previous Fri, Mon, Wed assignments will all be due Thur).

Videos
There will be a video, sometimes several videos, discussing the material in each section of the text (see the Weekly Syllabus for dates). These videos will be hosted on EDpuzzle, and you will access them through your EDpuzzle account (which you must create as described in the Course Tools). There will be check point questions interlaced throughout many of the videos; if you find these questions difficult, you should rewatch parts of the video and/or review parts of the text.

Here are a few important facts about the videos on EdPuzzle:

Course Exams

In the past, all course exams were given on the UC campus; I hope to return to this for the Final Exam and maybe even Exams 2,3 but probably not Exam 1. I will let you know more as we get closer to Exam 1.

Here are tentative dates for the exams; actual dates, times, and locations will be put here and on blackboard as soon as they are known. The first exam will cover Chapters 1,2. The second exam will cover Chapters 3,4. The third exam will cover Chapters 5,6,7. This will be made more precise. The final exam will be cumulative.

Course Content

 Here is a brief Course Syllabus.
  1. Systems of Linear Equations: Sections 1.1, 1.2, 1.3
  2. Matrix form of equations, Linear Independence: Sections 1.4, 1.5, 1.7
  3. Linear Transformations: Sections 1.8, 1.9
  4. Matrix Operations: Sections 2.1, 2.2, 2.3
  5. Vector Subspaces: Section 2.8, 2.9
  6. Determinants: Sections 3.1, 3.2
  7. Vector Spaces, Null Space, Column Space: Sections 2.8, 4.1, 4.2
  8. Linear Independence, Bases, Coordinates: Sections 4.3, 4.4
  9. Dimension, Rank, Change of Basis: Sections 2.9, 4.5, 4.6, 4.7
  10. Eigenvectors, Eigenvalues, Eigenbases (aka Diagonalization): Sections 5.1, 5.2, 5.3
  11. The matrix of an LT and eigenvectors of an LT: Section 5.4
  12. Orthogonality: Sections 6.1, 6.2
  13. Orthogonal Projections: Section 6.3
  14. Gram-Schmidt and QR Factorization: Section 6.4
  15. Least Squares: Section 6.5
  16. Spectral Theorem for Symmetric Matrices: Section 7.1
  17. Quadratic functions and Optimization: Sections 7.2, 7.3
  18. Singular Value Decomposition: Section 7.4
See the Weekly Syllabus for a more detailed day-by-day syllabus.

Here are some Learning Outcomes that a successful Linear Algebra student should be able to perform.
  1. Solve linear systems of equations using Gauss-Jordan elimination (i.e., elementary row operations).
  2. Perform common matrix operations with matrices containing real or complex numbers.
  3. Evaluate determinants, explain their properties, and use them in applications.
  4. Use both algebraic and geometric representations of Euclidean vectors in basic computations.
  5. Understand and use relationships between systems of linear equations, vector subspaces, linear transformations, and matrices.
  6. Work with abstract vector spaces.
  7. Compute eigenvalues and eigenspaces for matrices and when possible (orthogonally) diagonalize a matrix.
  8. Work with bases, including: verifying that a set is a basis, constructing bases, orthogonalizing bases, making a change of basis; and finding the dimension of a vector subspace. This means understanding the key concept of a linear combination, and the related ideas of linear independence versus linear dependence.
  9. Work with linear transformations, including finding the range, kernel, rank, nullity, and a matrix representation.
  10. Prove elementary theorems from linear algebra.
In this course students will develop the following transferable skills that apply to many different career settings in science, engineering, and business related fields:

Here is a more detailed syllabus along with some suggested homework. As the semester progresses, this will be modified as necessary. After each class, you should first try all of the Practice Problems in the sections covered. Then continue with the Suggested Exercises.

Weekly Syllabus
Week of Material Covered Suggested Exercises Remarks
Jan 11Sections 1.1, 1.2, 1.3 Section 1.1: 3, 7, 8, 11, 13, 17, 19-22, 25,27
Section 1.2: 1, 2, 5, 7, 9, 15, 16, 17, 19, 23, 29
Section 1.3: 5, 6, 9-14, 17, 25, 26 		Scan of initial suggested exercises
Jan 18Sections 1.4, 1.5 Section 1.4: 1-11 (odd), 12, 15, 17, 18, 21, 29, 30
Section 1.5: 3, 5, 9, 11, 13, 15, 17, 29-34		 No class Monday
M.L.King birthday
Jan 25Sections 1.7, 1.8, 1.9 Section 1.7: 1, 3, 5, 6, 7, 11, 13, 17, 23, 27, 32
Section 1.8: 5, 7, 8, 9, 11, 15, 16, 17, 19, 20, 31
Section 1.9: 3, 5, 7, 13, 15, 16, 17, 19, 25
Supplementary Exs Chpt 1: 1, 3, 5, 7, 11, 17, 21
Feb 1Sections 2.1, 2.2, 2.3 Section 2.1: 1, 3, 7, 11, 12, 13, 15, 17, 21, 27
Section 2.2: 1, 3, 7, 9, 15, 19, 21, 31, 33
Section 2.3: 1-8, 11, 15-19, 21, 33, 37		  
Feb 8 Review, Exam 1, Sections 2.8, 2.9 Section 2.8: 1, 2, 5, 7, 9, 11, 13, 17, 21, 23, 33
Section 2.9: 1, 3, 5, 9, 12, 13, 15, 16, 17, 21		 Exam 1 over Chpts 1, 2
Week of Material Covered Suggested Exercises Remarks
Feb 15 Sections 4.1, 4.2, 4.3 Section 4.1: 1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 21, 23, 31
Section 4.2: 1, 3, 5, 7, 11, 15, 17, 21, 25, 31, 33
Section 4.3: 1-6, 9, 11, 13, 15, 19, 21, 25		 President's Day Monday
Reading Day Tuesday
Feb 22 Sections 4.4, 4.5, 4.6 Section 4.4: 1, 3, 7, 9, 13, 14, 15, 17, 21, 25, 27, 32
Section 4.5: 1, 3, 5, 7, 11, 12, 13-19, 21, 23, 29
Section 4.6: 1, 3, 5, 7, 9, 13, 15, 17, 21, 23		  
Mar 1 Sections 4.7 (1.9, 5.4), 3.1 Section 4.7: 1, 3, 5, 7, 9, 11, 13
Section 1.9: 3, 5, 7, 13, 15, 16, 17, 19, 25
Section 5.4: 1, 3, 4, 5, 7, 8, 9, 11
Section 3.1: 3, 5, 9, 13, 27, 29, 37, 39
March 8 Section 3.2, Review, Exam 2 Section 3.2: 3, 7, 11, 15-20, 27, 29, 31-35, 39 Exam 2 over Chpts 3, 4
Week of Material Covered Suggested Exercises Remarks
March 15 Sections 5.1, 5.2, 5.3 Section 5.1: 1, 3, 15, 16, 25, 26, 27
Section 5.2: 3, 4, 7, 8, 9, 10, 17
Section 5.3: 1, 4, 5, 7-12, 17, 19, 23, 25, 23, 26, 31		 Spring Break ☺
March 22 Sections 5.4, 6.1 Section 5.4: 1, 3, 4, 5, 7, 8, 9, 11, 15, 19-22
Section 6.1: 1-10, 13, 17, 24, 25, 26, 27-31		 Reading Day
Wednesday
March 29 Sections 6.2, 6.3, 6.4 Section 6.2: 1, 5, 9, 10, 11-15, 17, 26, 27
Section 6.3: 1-15 (odd), 16, 19, 21, 22
Section 6.4: 3, 5, 9, 11, 15, 19-21		  
April 5 Sections 6.5, 7.1, 7.2 Section 6.5: 3, 5, 7, 11, 15, 25
Section 7.1: 1-27 (odd), 30
Section 7.2: 1,3-7, 9, 11, 13, 19		  
April 12 Review, Exam 3, Section 7.4 Section 7.4: 1, 3, 7, 9, 11, 13, 23, 25 Exam 3 over Chpts 5, 6, 7
April 19 Section 7.4, Review Section 7.4: 1, 3, 7, 9, 11, 13, 23, 25 End of Classes Wednesday
April  22Final Exam Thursday at 6:30pm Final Exams week

Section By Section

 Below is a section by section list of videos (and other files) that you should study. Please be sure to watch the videos (at least once) via edpuzzle—this is the only way that you get credit for watching these. In general, this section by section list will follow the Weekly Syllabus above. Each day will have a "master" pdf slide file (these are created using beamer, which is one way to use TeX/LaTeX/AMS-LaTex) that you can download and use (e.g. to take notes) when you watch the videos. These beamer files are used to create many of the videos, and sometimes several videos. There also may be related geogebra files and even some jpeg files; you can experiment with the geogebra files, and take notes on the jpegs.

First, here are some preparatory videos that serve as a Course Introduction. Here is a section by section list of videos (and other files) to help in your Linear Algebra adventure; I suggest watching and working thru these in the given order. Click on the triangle to see the files for a given chapter.

▼ CHAPTER 1

▼ CHAPTER 2

▼ CHAPTER 3

▼ CHAPTER 4

▼ CHAPTER 5

▼ CHAPTER 6

▼ CHAPTER 7

Course Help

 If you are seeking help, there are Graduate Student Teaching Assistants on duty at the MASS Center located in French Hall West room 2133. The MASS Center provides free services for the students in this course. During the times listed on their web page, students will be able to work collaboratively with each other under the guidance of a highly-trained tutor. No appointment is necessary for these tutoring sessions, but there are a limited number of seats available on a first-come, first-served in the MASS Center.

There is also one-on-one tutoring available where students will be able to work one-on-one with a qualified and trained peer tutor. Students may schedule individual tutoring appointments to improve their understanding of course materials and develop effective study strategies. The LAC also offers Academic Coaching. Academic Coaches are high achieving UC upperclassmen and graduate students who provide one-on-one support in order to encourage success-building practices and habits in students. Coaching is not course specific, but applicable to all majors and courses. LAC appointments are available Mon-Thurs 9am-8pm and Fri 9am-5pm. Students may schedule appointments online here or by contacting the LAC at (513) 556-3244. More information about the tutoring program is at the LAC website.

Perhaps the best way to get help is to ask your fellow classmates! In addition it is possible to hire a private tutor; see the MLC web page.

Finally, here is some friendly advice. I encourage you to get two notebooks for this course. Use one to write down class notes and problems that I work in class; do your homework problems in the other notebook. You will find it easier to study for exams if your class notes are not cluttered with your homework problems. I will go over as many problems as possible. However, mathematics is not a spectator sport; mathematical knowledge is not gained passively and you will not learn by osmosis; you must be an active participant in the learning process. This means that to learn the material you must work the problems yourself and practice every day. You must work lotsa problems, as many as you can. Don't be afraid to work some of the problems over and over, especially when you're studying for an exam. It is easy to fall behind; try to keep up with the course and seek help immediately if you have problems.

It is a excellent idea to go over your notes as soon as possible after class!

Regrading Policy

Mistakes are made in grading, especially when there is only one person responsible for grading all of your work. Sometimes these mistakes are in the student's favor, but unfortunately sometimes they are not. By following the procedure outlined below, you can have mistakes in the grading of your work corrected. Please be aware that just as it is likely that you will receive more points, it is also possible for you to actually lose points -- this generally happens to at least one person each term. Thus there are three possible outcomes of a regrade request: your score may remain the same or your score may increase or your score may decrease.

Note that partial credit is awarded only for work that is mostly correct except for one or two minor errors. You will not be given partial credit for attempting to solve a problem by the wrong method. Nor will you receive credit—even for a correct answer—if no supporting work is present.

Here is the Procedure to Follow for a Regrade Request. If you believe an error was made in grading your work, then you must appeal the grade in writing within one day of the day the work was returned to the class. A late request for regrading will automatically be denied. To have your work regraded, you must return it along with a clearly written note indicating the mistakes that you believe were made in grading. If your point totals were added incorrectly, simply indicate this on your regrade request. Otherwise, please provide the following information for each problem that you believe was graded incorrectly.

  1. The number of the problem to be regraded.
  2. The score you think you should receive.
  3. An explanation of why you think you deserve more points. This means that you should indicate which parts of your solution were graded incorrectly. You should be able to distinguish which part of your answer is correct and which part is incorrect. For example, you might say something like "I solved the problem correctly but forgot to multiply by 2 at the third step".
Note that no credit is given if you use the wrong method to solve a problem, even if your computations and/or your answers are correct. In order to provide the information asked for in part (2) above you may want to compare your solutions with the Answer Key which often will be available via blackboard.

Failure to provide any of the above information may result in your work not being regraded.

Lay's PowerPoint Files

Here are some of the powerpoint slides from the textbook.

GeoGebra

From google, geogebra is "A geometry package providing for both graphical and algebraic input....". Here's a link: geogebra.
Here are some geogebra files—with a brief description—that I have created; some of these appear in various videos, and there are also links to many geogebra files in the Section By Section lists.

Links

Here are some links to ....a text book, its solutions, another text book, a report.

University Information

Please see dates and deadlines for the last day to drop this class (with no entry to your academic record) and the last day to withdraw. These are official UC dates and something I have no control over. (However, you can in fact withdraw even after the stated date; you just cannot do so online but rather have to visit the A&S office.) If you withdraw from this course, I will be required to verify whether or not you minimally participated in the class. Although I will try my best to respond accurately, in the absence of any evidence to the contrary, I will state that you did not minimally participate. Ways for you to provide clear evidence of your presence in the class include turning in at least one homework assignment, taking at least one quiz, or taking at least one exam.

Academic Integrity Policy
The University Rules, including the Student Code of Conduct, and other documented policies of the department, college, and university related to academic integrity will be enforced. Any violation of these regulations, including acts of plagiarism or cheating, will be dealt with on an individual basis according to the severity of the misconduct.

Special Needs Policy
If you have any special needs related to your participation in this course, including identified visual impairment, hearing impairment, physical impairment, communication disorder, and/or specific learning disability that may influence your performance in this course, you should meet with the instructor to arrange for reasonable provisions to ensure an equitable opportunity to meet all the requirements of this course. At the discretion of the instructor, some accommodations may require prior approval by Disability Services.

If you expect to take the exams in the Disability Center, you must schedule to take them on the same date and to start at the same time as the rest of the class.

Except for a few courses, all mathematics classes satisfy the University Quantitative Reasoning Requirements. This course satisfies the QRR of UC's General Education program. This course was designed following the guidelines of the University of Cincinnati General Education Program. It satisfies, or partially satisfies, the Quantitative Reasoning distribution requirement. Moreover, of the five Baccalaureate Competencies, this course focuses on Critical Thinking, Effective Communication, and Information Literacy.

Counseling Services
Students have access to counseling and mental health care through the University Health Services (UHS), which provides both psychotherapy and psychiatric services. In addition, Counseling and Psychological Services (CAPS) provides professional counseling upon request; students may receive five free counseling sessions through CAPS without insurance. Students are encouraged to seek assistance for anxiety, depression, trauma/assault, adjustment to college life, interpersonal/relational difficulty, sexuality, family conflict, grief and loss, disordered eating and body image, alcohol and substance abuse, anger management, identity development and issues related to diversity, concerns associated with sexual orientation and spirituality concerns, as well as any other issue of concerns. After hours, students may call UHS at 513-556-2564 or CAPS Cares at 513-556-0648. For urgent physician consultation after-hours students may call 513-584-7777.

Title IX
Title IX is a federal civil rights law that prohibits discrimination on the basis of your actual or perceived sex, gender, gender identity, gender expression, or sexual orientation. Title IX also covers sexual violence, dating or domestic violence, and stalking. If you disclose a Title IX issue to me, I am required forward that information to the Title IX Office. They will follow up with you about how the University can take steps to address the impact on you and the community and make you aware of your rights and resources. Their priority is to make sure you are safe and successful here. You are not required to talk with the Title IX Office. If you would like to make a report of sex or gender-based discrimination, harassment or violence, or if you would like to know more about your rights and resources on campus, you can consult the website Title IX or contact the office at 556-3349.