MATH 2076 001, M W F 9:05-10 Edwards Room 6130

# Linear Algebra Spring 2024

Solutions: | Pdf's are posted on Canvas|

### Course Info

Instructor: Professor Wlodek Bryc, Office: 4316 French Hall (513) 556 4098, E-mail:
Office hours:
Textbook: David Lay, Linear Algebra and its applications, 6th edition. ISBN 9780135851159 This is available as an e-text as part of the MyLab Math access that comes as part of the course. You can access MyLab Math from our Canvas course (look in the menu on the left) and information you need about signing in is available in the first module (Getting Started) of our Canvas course
There are numerous free online resources for linear algebra. In the past, I used linear algebra textbook by Jim Hefferon. You may like a more interactive textbook, such as: Interactive Linear Algebra. Also there are numerous videos on YouTube that explain specific topics. I linked aomw vids on the eigenvectors and diagonalization under Remarks for suggested homework below. Dan Margalit, Joseph Rabinoff

• Exam 1: Monday, Feb 12. Material: Sections 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.8, 1.9, 2.1, 2.2, 2.3, 2.8, 2.9
• Exam 2: Monday, March 25. Material: Sections 3.1-3.3, 4.1-4.6, 5.1, 5.2, 5.3, 5.4
• Last day to withdraw: Friday, April 5
• Final Exam: Monday, April 22, 8AM-10AM
Material: TBA
Finals from previous years: TBP

Calculators Policy No books, notes, calculators and/or other electronic devices are allowed at any quiz, test, or exam. You are welcome to use them to check the accuracy of your work at home.

No makeups.

Homework/Quizzes 20%, Exams 25% each, Final 30%
${Course Average}=\frac15 Ave(Quiz+Hwk)+\frac12 Ave(Exams)+\frac{3}{10}Ave(Final)$

Grading scale: Grade cutoffs based on the Course Ave above are: 93% for A, 90% for A-, 80% for B-, 70% for C-, 60% for D-.

## Course description

Study of linear equations, matrices, Euclidean n-space and its subspaces, bases, dimension, coordinates, orthogonality, linear transformations, determinants, eigenvalues and eigenvectors, diagonalization.

## Topics and Section covered

1 Linear Equations in Linear Algebra
• 1.1 Systems of Linear Equations
• 1.2 Row Reduction and Echelon Forms
• 1.3 Vector Equations
• 1.4 The Matrix Equation $Ax = b$
• 1.5 Solution Sets of Linear Systems
• 1.7 Linear Independence
• 1.8 Introduction to Linear Transformations
• 1.9 The Matrix of a Linear Transformation

• 2 Matrix Algebra
• 2.1 Matrix Operations
• 2.2 The Inverse of a Matrix
• 2.3 Characterizations of Invertible Matrices
• 2.8 Subspaces of Rn
• 2.9 Dimension and Rank
• 3 Determinants
• 3.1 Introduction to Determinants
• 3.2 Properties of Determinants
• 4 Vector Spaces
• 4.1 Vector Spaces and Subspaces
• 4.2 Null Spaces, Column Spaces, Row Spaces, and Linear Transformations
• 4.3 Linearly Independent Sets; Bases
• 4.4 Coordinate Systems
• 4.5 The Dimension of a Vector Space
• 4.6 Change of Basis
• 5 Eigenvalues and Eigenvectors
• 5.1 Eigenvectors and Eigenvalues
• 5.2 The Characteristic Equation
• 5.3 Diagonalization
• 5.4 Eigenvectors and Linear Transformations
• 6 Orthogonality and Least Squares
• 6.1 Inner Product, Length, and Orthogonality
• 6.2 Orthogonal Sets
• 6.3 Orthogonal Projections
• 6.4 The Gramâ€“Schmidt Process, QR
• 6.5 Least-Squares Problems
• 6.7 Inner Product Spaces

• 7 Symmetric Matrices and Quadratic Forms
• 7.1 Diagonalization of Symmetric Matrices
• 7.4 The Singular Value Decomposition

## Suggested homework

After each class, you should first try all Practice Problems for the section covered in class. Then continue with the assigned homework or with the suggested exercises. Once you work out enough examples by hand and become comfortable with how things work, you may speed up calculations by using software: Matlab, Mathematica, Maple, Sage (free) or online tools like WolframApha, https://matrixcalc.org/ or SymboLab  Week of Material Suggested Exercises Remarks Jan 8 Sections 1.1, 1.2, 1.3 Section 1.1: 3, 7, 8, 11, 13, 17, 19, 23-26, 35 Section 1.2: 2, 7, 9, 15, 19, 21, 23, 45 Section 1.3: 5, 6, 9-14, 17, 18 scan of hwk(Sect 1.1+1.2 PDF) Jan 15 Sections 1.4, 1.5 Section 1.4: 1-4, 6, 7, 11, 12, 15 Section 1.5: 5, 8, 11, 17, 18, 23, 25 no class Jan 15 Jan 22 Sections 1.7, 1.8, 1.9 Section 1.7: 1-4, 5, 6, 8, 11, 12 Section 1.8: 2-6, 9, 11, 19, 20, 31 Section 1.9: 5, 6, 15, 16, 17, 18, 22 Jan 29 Sections 2.1, 2.2, 2.3 Section 2.1: 1, 2, 6, 7, 9, 11, 12, 13, 19-22, 25 Section 2.2: 1, 4, 6, 7, 41, 43, 45, 48 Section 2.3: 1-8, 11-20 Feb 5 Sections 2.8. 2.9. Review for Exam 1 over Chapters 1, and 2 Section 2.8: 1-4, 5, 6, 11, 12, 31-34 Section 2.9: 1, 3-6, 11-13 Feb 12 Exam 1 (Monday), Sections 3.1, 3.2, 3.3 We+Fri. Section 3.1: 3, 4, 13, 14, 25-30, 38 Section 3.2: 7, 8, 11, 12, 15-20, 23, 24, 29, 37, 45 Section 3.3: 16-18, 19, 20, 21, 23 Week of Material Suggested Exercises Remarks Feb 19 Sections 4.1, 4.2, 4.3 Section 4.1: 1, 2, 9, 10, 15, 17, 19, 20 Section 4.2: 1, 3-6, 15, 17-20, 24 43-45 Section 4.3: 1-6, 9, 10, 11, 13, 14, 15, 16, 20, 43, 44, 47 (use derivatives!), 48 for three functions $\{1, \cos t, \cos^2t\}$ Feb 26 Sections 4.4, 4.5. 4.6 Section 4.4: 2, 3, 7, 8, 13, 14, 31-36, 39 Section 4.5: 3, 4, 10, 11-15, 27-30, 54 with fewer functions $\mathcal{B}=\{1, \cos t, \cos^2t\}$, $\mathcal{C}=\{1,\cos t, \cos 2t\}$. Section 4.6: 1, 2, 4, 5, 6, 8, 15, 16 March 4 Sections 5.1, 5.2, 5.3 Section 5.1: 1, 3, 15, 16, 17, 18, 33, 34, 35 Section 5.2: 3, 4, 7, 8 (find the eigenvalues and the eigenvectors), 9, 10, 17, 35 Section 5.3: 4, 5, 7-12, 19, 20 Eigenvectors example by Trevor (UC) Refresher: NancyPi divides polynomials Diagonalization $A=PDP^{-1}$: Theory by Dr. Herron (UC) and by Trevor Examples: | Trevor (UC) | Computing $A^n$ | March 11 Spring break Week of Material Suggested Exercises Remarks March 18 Sections 5.3, 5.4, Review Section 5.4: 1, 3, 4, 5, 7, 8, 9, 13, 21-25 March 24 Exam 2 (Monday), Sections 6.1, 6.2 (We+Fri) Section 6.1: 1-8, 12, 14, 33, 34 (find a basis in this subspace), 35-39 Section 6.2: 9, 10, 12-15 Apr 1 Sections 6.3, 6.4, 6.5 or 6.7 Section 6.3: 1, 3-6, 7, 8, 11, 13, 15, 19 Section 6.4: 3, 4, 9-12, 16 Section 6.5: 1-6 Section 6.7: 3, 5, 7, 31 Apr 8 Sections 7.1, 7.2 Section 7.1: 7-12, 13-22, 23, 24 Section 7.2: 3-6, 9-14 Apr 15 Section 7.4, Review Section 7.4: 1, 3, 7, 14 Apr 22 Final Exam Final Exams week

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.

### Tutoring

Students may schedule appointments online at the Learning Commons site: https://www.uc.edu/learningcommons.html or by contacting the Learning Commons at (513) 556-3244.

Drop-in tutoring is offered for this course by the Math and Science Support (MASS) Center. You can find the schedule and more information about the MASS Center at https://www.uc.edu/learningcommons/masscenter.html.

## Safety

UC Night Ride is a student organization that provides any UC student, faculty, or staff member transportation to the neighborhoods around campus after dark. The phone number is 513-556-RIDE (7433)

For public health updates see https://uc.edu/publichealth.html

## Religious Accommodations

Ohio law and the Universityâ€™s Student Religious Accommodations for Courses Policy 1.3.7 permits a student, upon request, to be absent for reasons of faith or religious or spiritual belief system or participate in organized activities conducted under the auspices of a religious denomination, church, or other religious or spiritual organization and/or to receive alternative accommodations with regard to examinations and other course requirements due to an absence permitted for the above-described reasons. Not later than fourteen days after the first day of instruction in the course, a student should provide the instructor with written notice of the specific dates for which the student requests alternative accommodations. For additional information about this policy, please contact the Executive Director of the Office of Equal Opportunity and Access at (513) 556-5503 or oeohelp@UCMAIL.UC.EDU.

#### Syllabus is subject to change

This syllabus may be updated with additional information as it becomes available. Please refresh your browser to make sure that the updates are visible. Revision date appears at the top and at the bottom of this page.