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Penn State, University Park, Fall 2022

MATH 220H — MATRICES

Mon Wed Fri from 09:05 to 09:55 in Willard Building 271

The instructor reserves the right to make changes to this syllabus during the semester.

Instructor

Professor Mathieu Stiénon

Information and Communications Technology

All electronic devices must be silenced during lectures. Disrupting class in any way will not be tolerated.

We will use Canvas to share course materials, Gradescope to administer assignments, Teams for instant messaging, and Zoom for virtual office hours.

IT support is available 24/7.

     
Canvas iOS android
Gradescope iOS android
Teams iOS android
Zoom iOS android

Gradescope

Your Gradescope account will be created automatically the first time you click on the “Gradescope” tab in Canvas.

Using Gradescope as a Student

Instant Messaging

We will use Microsoft Teams for text-based communication outside the classroom.

Depending on the issue at hand, we will use either the team channel or the private chat features of Teams. Watch this video to learn how team channels (i.e. public conversations) work. Watch this other video to learn how chats (i.e. private one-on-one or group conversations) work.

Instructions for adding yourself to our team:

  1. Copy the team code you’ll find in the ‘Course Administration’ module in Canvas.
  2. Login to Office 365 with your PSU email address and password.
  3. Click on the Teams application icon.
  4. In the left menu bar of Teams, click on the Teams icon.
  5. Click on ‘Join or Create a Team’ (located in the bottom, left corner of the screen).
  6. In the box for ‘Join a Team with a Code’, paste the Team Code.
  7. Click the ‘Join Team’ button.

For your convenience, I encourage you to install the Teams app on your phone and/or tablet as well.

Please take a moment now to download the app and join the team as I will share important information with the class through Teams rather than through Canvas announcements and your Canvas inbox. Do it now!

Virtual Office Hours

Tentatively scheduled on Wednesdays from 17:00 to 18:00.

Check the ‘course administration module’ in Canvas for the day and time of virtual office hours.

We will use Zoom in conjunction with a shared online whiteboard. You will find instructions and links for joining the Zoom meeting and the shared online whiteboard in the ‘Course Administration’ module in Canvas. You must be logged into the Zoom app with your PSU credentials to be allowed into the meeting.

Disrupting virtual office hours in any way will not be tolerated. Refusal to comply with University policies is a violation of the Code of Conduct. Students may face disciplinary action for Code of Conduct violations.

“David Lay, Linear Algebra and its Applications, Pearson”

The sixth edition is the current edition. However, the content is virtually the same in all editions. The only difference I noticed is that the exercises are changed from one edition to the next. Most problems are exactly the same but the numerical values in each problem are changed. Thus, no matter which edition you get, you will be solving the same problems but with different numbers. If I were you, I’d buy the cheapest used copy I could find that is still in good physical shape.

Be smart: buy a past edition of the textbook and save money.

Course Description

Honors course in systems of linear equations; matrix algebra; eigenvalues and eigenvectors.

This course is intended as an introduction to linear algebra with a focus on solving systems for linear equations. Topics include systems of linear equations, row reduction and echelon forms, linear independence, introduction to linear transformations, matrix operations, inverse matrices, dimension and rank, determinants, eigenvalues, eigenvectors, diagonalization, and orthogonality.

In contrast to the non-honors version of this course, the honors version is typically more theoretical and will often include more sophisticated problems. Moreover, certain topics are often discussed in more depth and are sometimes expanded to include applications which are not visited in the non-honors version of the course.

Prerequisites

MATH 110 , MATH 140 , or MATH 140H

Course Objectives

1.1 Systems of Linear Equations

  1. Use elementary row operations to solve systems of linear equations.
  2. Determine if a system of linear equations is consistent.
  3. Determine the conditions for which a linear system is consistent
  4. Determine the validity of statements about systems of linear equations, row operations, or matrices.

1.2 Row Reduction and Echelon Forms

  1. Identify matrices in echelon form and reduced echelon form.
  2. Row reduce matrices to reduced echelon form.
  3. Find the general solution to a system with a given augmented matrix.
  4. Determine if a solution is consistent given a description of the corresponding coefficient matrix.
  5. Determine the conditions for which a linear system has specified types of solutions.
  6. Determine the validity of statements about row reduction and echelonforms.

1.3 Vector Equations

  1. Compute sums and scalar products of vectors, both algebraically and geometrically.
  2. Convert between vector equations and systems of equations.
  3. Determine if a vector is a linear combination of other vectors.
  4. Characterize the span of a set of vectors algebraically or geometrically.
  5. Determine the validity of statements about vectors and vector equations.

1.4 The Matrix Equation $A\vec{x}=\vec{b}$

  1. Compute the product of a matrix and a vector.
  2. Convert between matrix equations, vector equations, and systems of equations.
  3. Solve matrix equations using augmented matrices.
  4. Characterize the span of the column vectors of a matrix.
  5. Determine whether a matrix equation has no solution, one solution, or infinitely many solutions.
  6. Determine the validity of statements about vector equations and matrix equations.

1.5 Solution Sets of Linear Systems

  1. Determine if a system of equations has a nontrivial solution.
  2. Solve a system of equations or a matrix equation and write the solution in parametric form.
  3. Describe the solution sets of systems of equations geometrically.
  4. Determine the validity of statements about solution sets of linear equations.

1.7 Linear Independence

  1. Determine if a set of vectors is linearly independent and determine if a vector is in a given span.
  2. Determine conditions for which vectors are linearly independent or have a given span.
  3. Determine the validity of statements about linear independence.

1.8 Introduction to Linear Transformations

  1. Algebraically find the image of a given vector under a linear transformation.
  2. Given a linear transformation $T:\mathbb{R}^n\to\mathbb{R}^m$ and a vector $\vec{b}\in\mathbb{R}^m$, find all vectors of $\mathbb{R}^n$ having $\vec{b}$ as image under $T$.
  3. Determine the conditions for which a linear transformation has a given domain and codomain.
  4. Determine if a vector is in the range of a linear transformation.
  5. Geometrically describe the image of a vector under a linear transformation.
  6. Use the linearity of transformations to find the images of vectors under the transformation.
  7. Determine the validity of statements about lineartransformations.

1.9 The Matrix of a Linear Transformation

  1. Find the standard matrix of a linear transformation.
  2. Find vectors whose images under a linear transformation are given.
  3. Determine the validity of statements about properties of linear transformations.
  4. Determine if linear transformations are one-to-one or onto.

2.1 Matrix Operations

  1. Compute sums, products, and scalar products of matrices.
  2. Find values of matrices such that products of matrices have given properties.
  3. Determine the validity of statements about matrix operations.

2.2 The Inverse of a Matrix

  1. Use the row reduction algorithm to figure out whether a given square matrix is invertible or not.
  2. Use elementary row operations to obtain the inverse of an (invertible) square matrix.
  3. Use matrix inversion to solve a linear system (with invertible coefficient matrix).
  4. Determine the validity of statements about inverses of matrices.
  5. Solve equations involving invertible matrices.

2.3 Characterizations of Invertible Matrices

  1. Determine if a given matrix is invertible.
  2. Solve problems involving transformations and their matrices.

2.8 Subspaces of $\mathbb{R}^n$

  1. Explain why a set is not a subspace of $\mathbb{R}^n$.
  2. Find a vector in a vector space or determine if a vector is in a vector space.
  3. Find the dimension of the null and column spaces of a matrix.
  4. Determine if a set of vectors is a basis for $\mathbb{R}^2$ or $\mathbb{R}^3$.
  5. Determine the validity of statements about subspaces of $\mathbb{R}^n$.
  6. Find bases for the null and column spaces of a matrix.

2.9 Dimension and Rank

  1. Find a coordinate vector in a subspace
  2. Find a basis of a subspace and determine the dimension.
  3. Solve conceptual problems involving dimension and rank.
  4. Determine the validity of statements about dimension and rank.

3.1 Introduction to Determinants

  1. Compute determinants using cofactor expansions.
  2. Determine the effect of elementary row operations on a determinant.
  3. Compute determinants of elementary matrices.
  4. Calculate determinants of scalar multiples of matrices.

3.2 Properties of Determinants

  1. Identify properties of determinants
  2. Find determinants by row reduction to echelon forms.
  3. Use properties of determinants to evaluate determinants.
  4. Prove properties of determinants.
  5. Use determinants to determine if a matrix is invertible or a set of vectors is linearly independent.
  6. Determine the validity ofstatements about properties of determinants.

3.3 Cramer’s Rule, Area and Volume

  1. Use Cramer’s rule to compute the solutions of systems of equations.
  2. Find the area of a parallelogram or volume of a parallelepiped using matrix determinants.

5.1 Eigenvectors and Eigenvalues

  1. Determine if a vector or number is an eigenvector or eigenvalue of a given matrix.
  2. Find a basis for the eigenspace corresponding to an eigenvalue.
  3. Find the eigenvalues of matrices.
  4. Determine the validity of statements about eigenvectors and eigenvalues.

5.2 The Characteristic Equation

  1. Find the characteristic polynomial and eigenvalues of a 2x2 matrix.
  2. Find the characteristic polynomial of a 3x3 matrix.
  3. Find the eigenvalues of triangular matrices.

5.3 Diagonalization

  1. Compute $A^k$ for $A=PDP^{-1}$.
  2. Use the diagonalization theorem to find the eigenvalues of a matrix and a basis for each eigenspace.
  3. Diagonalize matrices.
  4. Determine the validity of statements about diagonalization.
  5. Determine whether matrices are diagonalizable.
  6. Answer conceptual questions about diagonalization of matrices.

6.1 Inner Product, Length and Orthogonality

  1. Perform operations on vectors using inner products.
  2. Find a unit vector.
  3. Find the distance between two vectors.
  4. Determine whether two vectors are orthogonal.

6.2 Orthogonal Sets

  1. Determine whether a set of vectors is orthogonal.
  2. Express a given vector as a linear combination of vectors in an orthogonal basis.
  3. Find an orthogonal projection onto a line through a given vector and the origin.
  4. Write a vector as a sumof two orthogonal vectors.
  5. Find the distance between a vector and a line through the origin.
  6. Determine whether a set of vectors is orthonormal.
  7. Determine the validity of statements about orthogonal sets or matrices and orthogonal projections.

6.3 Orthogonal projections

  1. Find the orthogonal decomposition of a vector.
  2. Find an orthogonal projection onto a given subspace.
  3. Use the best approximation theorem to find a closest point or a distance.
  4. Construct a vector that is orthogonal to a given orthogonal set.
  5. Determine the validity of statements about orthogonal projections onto subspaces.

6.4 The Gram-Schmidt Process

  1. Use the Gram-Schmidt process to produce an orthogonal basis and find an orthonormal basis.

7.1 Diagonalization of Symmetric Matrices

  1. Identify symmetric matrices.
  2. Identify orthogonal matrices and find their inverse.
  3. Orthogonally diagonalize a matrix.
  4. Determine the validity of statements about diagonalization of symmetric matrices

Learning Objectives

Upon successful completion of MATH 220, the student should be able to:

  1. Know what is meant by a system of linear equations (or linear system) and its solution set.
  2. Know how to write down the coefficient matrix and augmented matrix of a linear system.
  3. Use elementary row operations to reduce matrices to echelon forms.
  4. Make use of echelon forms in finding the solution sets of linear systems.
  5. Know how to manipulate with vectors in Euclidean space.
  6. Understand the meaning of linear independence/dependence and span.
  7. Interpret linear systems as vector equations.
  8. Define matrix vector product and be able to interpret linear systems as matrix equations.
  9. Write the general solution of linear systems in parametric vector form.
  10. Understand the relation between the solution set of a consistent inhomogeneous linear system and its associated homogeneous equation.
  11. Determine whether sets of vectors are linearly independent or dependent.
  12. Know what is meant by a linear transformation between Euclidean spaces.
  13. Determine the standard matrix of a linear transformation.
  14. Give the geometric description of some matrices.
  15. Understand the notion of one-to-one mapping and onto mapping.
  16. Know how to scale a matrix, take the transpose of a matrix, and how to add and multiply matrices.
  17. Know what is meant by an invertible matrix.
  18. Know how to compute the inverse of a matrix, if it exists.
  19. Understand the various characterizations of an invertible matrix.
  20. Determine if a given subset of a Euclidean space is a subspace.
  21. Know what is the column space and nullspace of a matrix and how to determine these spaces.
  22. Find a basis of a subspace of a Euclidean space.
  23. Define the concept of dimension and how to use the rank plus nullity theorem.
  24. Know the recursive definition of determinants.
  25. Make use of the properties of determinants in their calculations.
  26. Find eigenvalues and eigenvectors of square matrices.
  27. Diagonalize square matrices, whenever possible.
  28. Compute the matrix of a linear transformation relative to given bases.
  29. Compute the inner product of vectors, lengths of vectors, and determine if vectors are orthogonal.
  30. Know what is meant by an orthogonal set, orthogonal basis and orthogonal matrix.
  31. Find the orthogonal projection of a vector onto a subspace.
  32. Find an orthogonal basis using the Gram-Schmidt process.
  33. Determine the least-squares solutions of linear systems.
  34. Orthogonally diagonalize symmetric matrices.
  35. Know how to eliminate cross-product terms in quadratic forms.

Weekly Problem Sets

Problem sets will be assigned weekly on Friday and are due one week later. All problem sets will have equal weight.

Discussion of problem sets with others is allowed, but each student should write his or her own solutions independently. You should definitely never share work you are submitting for credit with other students.

Once you have completed the assignment for the week, you will scan your work and submit the resulting PDF file through Gradescope.

Submit your work well ahead of the deadline. Assignments submitted late will automatically be marked as such in Gradescope and the Canvas grade-book. If you submit too many assignments late, I reserve the right to apply a penalty to your grade.

Problem Sets will lock 48 hours after their respective due dates and late submission will be disabled at that time.

Students who have a valid and documented reason for not submitting problem solutions on time may request to be excused for that assignment. However, there will be no makeup assignments.

Scanning Apps for Your Phones and Tablets

The easiest way to scan and submit your work is to use the Gradescope app for your phone.

The mobile apps for Microsoft OneDrive and Google Drive allow you to scan your completed assignments, convert them into PDF files, and store them in the cloud. As a PSU student, you have virtually unlimited cloud storage on OneDrive and Google Drive — you must be logged into the app with your PSU credentials.

     
Gradescope iOS android
OneDrive iOS android
Google Drive iOS android

Instructions for Assignment Submission

  1. Download the PDF of the assignment from Gradescope.
  2. Solve the problems using a black pen either on the printout or on blank sheets of white letter-size printer paper. Show all your work and explain your work briefly in full sentences. Final answers without supporting work and explanations will not receive credit. Write cleanly! If your work is not easily readable and intelligible, you will loose points!
  3. Scan your work, and submit it as one single PDF file in Gradescope. If you’d rather work directly on your pen-enabled tablet, make sure to submit all layers of the PDF document.
  4. In Gradescope, once you have uploaded your PDF file, you must match the pages of your document to the questions of the assignment. Otherwise, your submission will be considered incomplete and will receive a zero score.
  5. Check that your submission is complete and that you have submitted all pages. Unreadable work will result in a zero score.
  6. Assignments dropped in the instructor’s mailbox, sent to his email address, slipped under his office door, or submitted in Teams chat are not acceptable, will not be acknowledged, will be ignored, and will not be graded. You must submit your work through Gradescope.

Examinations

Students in this class are expected to complete the examinations on their own, and to write their answers in their own words. Students are not to obtain exam answers from any other person and present them as their own. Students who present other people’s work as their own may receive a zero score on the examination and an F or XF grade for the course.

No books, notes, or calculators may be used on the examinations.

Should I have any reason to believe that you may have submitted work which is not your own, I will request that you meet with me through videoconferencing, I will have you explain your work to me and I will ask you follow-up questions to test your understanding of the material. If you are not able to answer satisfactorily, I reserve the right to reduce your score on the examination, possibly down to zero. Academic integrity matters!

Warning: All University and College policies regarding academic integrity/academic dishonesty apply to this course and to the students enrolled in this course. Academic dishonesty could result in a transcript notation indicating failure due to academic misconduct.

Midterm Examinations

Two midterm examinations will be given in class on Friday September 30, 2022 and Friday October 28, 2022.

If there is a valid and documented reason for not being available during the regular mid-term examination time-window, such as classes or other official university activities or illness, a student may arrange with the instructor to take a makeup exam.

You need to IM (Teams) your instructor 48 hours prior to the exam regarding conflicts, and within 24 hours of missing the exam due to illness. If you miss the mid-term exam without an official excuse (such as illness or official university business), you may be allowed to take a makeup exam, but with an automatic 15% deduction from the grade. To avoid this deduction, you must notify your instructor via IM (Teams) with your official excuse, within 24 hours of the missed exam.

Forgetting the date or time of an examination is not a valid excuse.

Final Examination

A comprehensive final examination will be given during the final examination period (December 12-16, 2022).

The final examination may be scheduled on any day during the final examination period. Do not plan to leave University Park until after Friday, December 16, 2022. Students may access their final exam schedules Monday, September 26, 2022 through their LionPATH account.

Notification of conflicts is given on the student’s final exam schedule. There are two types of conflict final examinations: direct and overload. Direct conflicts are two examinations scheduled at the same time. Overload examinations are defined as three or more examinations scheduled in consecutive time periods or within one calendar day. Students may elect to take the three or more examinations on the same day if they wish or request a conflict final examination. A student must take action to request a conflict exam through LionPATH between September 26 and October 16, 2022. Conflict final examinations cannot be scheduled through the Mathematics department.

Students who miss or cannot take the final examination due to a valid and documented reason, such as illness, may be allowed to take a makeup final examination at the beginning of the next semester. Personal business, such as travel, employment, weddings, graduations, or attendance at public events such as concerts and sporting events is not a valid excuse. Forgetting the date or time of the examination is not a valid excuse. All such makeup examinations must be arranged through the instructor, and students in such a situation should contact their instructors within 24 hours of the scheduled final examination.

Students who miss the final exam without a valid reason might be allowed to take a makeup exam but a 15% penalty will be imposed. Students who have taken the original final examination are not permitted to take a makeup examination.

Grades

Assignments will be weighted as follows:

Type of Assignment Weight
Weekly Problem Sets 40%
First Midterm Examination 15%
Second Midterm Examination 15%
Final Examination 30%

Course letter grades will be assigned as follows:

100% ⩾ A ⩾ 93% > A- ⩾ 90% > B+ ⩾ 87% > B ⩾ 83% > B- ⩾ 80% > C+ ⩾ 77% > C ⩾ 70% > D ⩾ 60% > F ⩾ 0%

The unavoidable consequence is that some students will be ‘just a point’ away from the next higher or lower grade. For reasons of fairness, the policy in this course is to NOT adjust individual grades in such circumstances.

Your grade will be based exclusively on the problem sets, the midterm examinations, and the final examination. There is no ‘extra-credit’ work.

Deferred Grades

Students who are currently passing a course but are unable to complete the course because of illness or emergency may be granted a deferred grade which will allow the student to complete the course within the first several weeks of the following semester. Note that deferred grades are limited to those students who can verify and document a valid reason for not being able to take the final examination. For more information see Deferred Grades.

Late-Drop

Students may add/drop a course without academic penalty within the first six calendar days of the semester. A student may late drop a course within the first twelve weeks of the semester but accrues late drop credits equal to the number of credits in the dropped course. A baccalaureate student is limited to 16 late drop credits. The late drop deadline for Fall 2022 is Friday, November 11, 2022 at 23:59 (Eastern Time).

Tutoring

Free mathematics tutoring is available at Penn State Learning. They offer both online and in-person options.

For more help, the Department of Mathematics maintains a list of private tutors.

Counseling and Psychological Services

Many students at Penn State face personal challenges or have psychological needs that may interfere with their academic progress, social development, or emotional well-being. The university offers a variety of confidential services to help you through difficult times, including individual and group counseling, crisis intervention, consultations, online chats, and mental health screenings. These services are provided by staff who welcome all students and embrace a philosophy respectful of students’ cultural and religious backgrounds, and sensitive to differences in race, ability, gender identity and sexual orientation.

The National Suicide Prevention Lifeline provides 24/7, free and confidential support for people in distress, prevention and crisis resources for you or your loved ones, and best practices for professionals.

Academic Integrity

Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University’s Code of Conduct states that all students should act with personal integrity, respect other students’ dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts. Academic integrity includes a commitment not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others. In order to ensure all students have a fair and equal opportunity to succeed in this course, the Mathematics Department is committed to enforcing the University’s academic integrity policy. Below is a description of academic misconduct and the department’s responsibilities when misconduct is suspected.

Academic Misconduct

In this course, academic misconduct includes, but is not limited to:

When Academic Misconduct is Suspected

If a student is suspected of academic misconduct, the instructor’s duties are to:

Note that a student’s refusal to meet with the instructor or respond to the charges within a reasonable period of time is construed as acceptance of the allegation and proposed sanctions.

Once the Academic Integrity form has been accepted or contested by the student, it is sent to the College’s Academic Integrity Committee for adjudication. A student cannot drop or withdraw from the course during the adjudication process.

Sanctions

If a student accepts an academic misconduct allegation, or if (s)he is found guilty during adjudication, probable sanctions include:

Additional sanctions might include:

In addition, the student will be unable to drop or withdraw from the course.

Please see the Eberly College of Science Academic Integrity homepage for additional information and procedures. Also see the Code of Ethics for Engineers published by the National Society of Professional Engineers.

Students with Disabilities

Penn State welcomes students with disabilities into the University’s educational programs. If you have a disability-related need for reasonable academic adjustments in this course, contact Student Disability Resources at 814-863-1807 (V/TTY). For further information, please visit the Student Disability Resources web site. In order to receive consideration for accommodations, you must contact SDR and provide documentation (see the documentation guidelines at the Student Disability Resources web site). If the documentation supports your request for reasonable accommodations, SDR will provide you with an accommodation letter identifying appropriate academic adjustments. Please share this letter with your instructors and discuss the accommodations with them as early in your courses as possible. You must follow this process for every semester that you request accommodations.

Code of Mutual Respect and Cooperation

The Eberly College of Science Code of Mutual Respect and Cooperation pertains to all members of the college community; faculty, staff, and students. The Code of Mutual Respect and Cooperation was developed to embody the values that we hope our faculty, staff, and students possess, consistent with the aspirational goals expressed in the Penn State Principles. The University is strongly committed to freedom of expression, and consequently, the Code does not constitute University or College policy, and is not intended to interfere in any way with an individual’s academic or personal freedoms. We hope, however, that individuals will voluntarily endorse the 12 principles set forth in the Code, thereby helping us make the Eberly College of Science a place where every individual feels respected and valued, as well as challenged and rewarded.

Educational Equity

The Office of the Vice Provost for Educational Equity serves as a catalyst and advocate for Penn State’s diversity and inclusion initiatives. Educational Equity’s vision is a Penn State community that is an inclusive and welcoming environment for all. If you wish to learn more or if you wish to report bias, please visit the Educational Equity website.

All course materials students receive or to which students have online access are protected by copyright laws. Students may use course materials and make copies for their own use as needed, but unauthorized distribution and/or uploading of materials without the instructor’s express permission is strictly prohibited. University Policy AD 40, the University Policy on Recording of Classroom Activities and Note Taking Services, addresses this issue. Students who engage in the unauthorized distribution of copyrighted materials may be held in violation of the University’s Code of Conduct and/or liable under Federal and State laws.

A rising trend across the University is the posting and/or retrieval of material from course-share sites. Generally speaking, the uploading of materials to a course-share site is viewed as an Intellectual Property violation, and the downloading and use of materials from a course-share site could be a violation of academic integrity. If you have questions regarding the specific use of such a site, seek clarification directly from your instructor.

Student Class Recordings

Students are not allowed to record class sessions without permission.

According to University Policy, students must get express permission from their instructor to record class sessions. Screenshots showing instructors and students are considered recordings. Even if permission is granted, student-initiated recordings must be used only for educational purposes for the students enrolled in the initiating student’s class. Recordings may be used only during the period in which the student is enrolled in the class. Authorized student-initiated recordings may not be posted or shared in any fashion outside of the class, including online or through other media, without the express written consent of the course instructor or appropriate University administrator. Students who engage in the unauthorized distribution of class recordings may be held in violation of the University’s Code of Conduct, and/or liable under Federal and State laws.