MATH 2076-005  TuTh 3:30PM - 4:50PM  French Hall 4221

Linear Algebra Spring 2026

Course Info

Instructor: Professor Wlodek Bryc, Office: 4316 French Hall Office hours Tel: (513) 556 4098, E-mail:

Textbook: Lay, McDonald. Linear Algebra and its applications, 6th edition
This is an e-text available as part of the Follett Willo access that comes with the course. Go to Pearson Course Resources in the Follett Willo 1.3 module on Canvas.

Exams and important dates

Exam 1: Thursday, 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: Thursday, March 26. Material: Sections 3.1-3.3, 4.1-4.6, 5.1, 5.2, 5.3, 5.4
Last day to withdraw: Friday, April 10
Exam 3: Thursday, April 23 Material: Sections 6.1-6.5, 7.1, 7.2
This section of linear algebra (MATH 2076-005) will have no Final Exam

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.

Grading

Homework/Quizzes 5%+20%=25%, Exams 25% each

Grading scale: Grade cutoffs based on the Course Ave are: 93% for A, 90% for A-, 80% for B-, 70% for C-, 60% for D-.
To estimate your current standing in the course, please look at your Total under Grades on Canvas.

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

1.1 Systems of Linear Equations
1.2 Row Reduction and Echelon Forms
1.3 Vector Equations
1.4 The Matrix Equation $A\vec x = \vec b$
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 $\mathbb{R}^n$
- 2.9 Dimension and Rank
- Exam 1
3 Determinants
- 3.1 Introduction to Determinants
- 3.2 Properties of Determinants
- 3.3 Cramer's rule, Volume, and Linear Transformations
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
- Exam 2
6 Orthogonality and Least Squares
- 6.1 Inner Product, Length, and Orthogonality
- 6.2 Orthogonal Sets
- 6.3 Orthogonal Projections
- 6.4 The Gram-Schmidt Process, QR
- 6.5 Least-Squares Problems

7 Symmetric Matrices and Quadratic Forms

7.1 Diagonalization of Symmetric Matrices
7.2 Quadratic Forms

7.4 The Singular Value Decomposition
Exam 3

Suggested exercises

After each class, you should first try the Practice Problems for the sections covered in class. Then do the assigned homework and 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 13+15	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 20+22	Sections 1.4, 1.5, 1.7	Section 1.4: 1-4, 6, 7, 11, 12, 15 Section 1.5: 5, 8, 11, 17, 18, 23, 25 Section 1.7: 1-4, 5, 6, 8, 11, 12
Jan 27+29	Sections 1.8, 1.9, 2.1	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 Section 2.1: 1, 2, 6, 7, 9, 11, 12, 13, 19-22, 25
Feb 3+5	Sections 2.2, 2.3, 2.8	Section 2.2: 1, 4, 6, 7, 41, 43, 45, 48 Section 2.3: 1-8, 11-20 Section 2.8: 1-4, 5, 6, 11, 12, 31-34
Feb 10+12	Section 2.9. Exam 1	Section 2.9: 1, 3-6, 11-13
Week of	Material	Suggested Exercises	*Remarks*
Feb 17+19	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
Feb 24+26	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 the derivatives!), 48 but only for three functions: $\{1, \cos t, \cos^2t\}$
March 3+5	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 10+12	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 Trevor Examples: \| Trevor (UC) \| Computing $A^n$ \|
March 16-22		Spring break
March 24+26	Section 5.4, Exam 2	Section 5.4: 1, 3, 4, 5, 7, 8, 9, 13, 21-25
Week of	Material	Suggested Exercises	*Remarks*
Mar 31 +Apr 2	Sections 6.1, 6.2, 6.3	Section 6.1: 1-8, 12, 14, 33, 34 (find a basis in this subspace), 35-39 Section 6.2: 9, 10, 12-15 Section 6.3: 1, 3-6, 7, 8, 11, 13, 15, 19
Apr 7+9	Sections 6.4, 6.5, 7.1,	Section 6.4: 3, 4, 9-12, 16 Section 6.5: 1-6 Section 7.1: 7-12, 13-22, 23, 24
Apr 14+16	Sections 7.2, 7.3(?) 7.4(?)	Section 7.2: 3-6, 9-14 Section 7.3: 1, 3, 5, 7 Section 7.4: 1, 3, 7, 14
Apr 21+23	Review, Exam 3

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.

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.

Safety

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

For weather closures, see UC Status page: http://www.uc.edu/alert.html

Student Religious Accommodations

This course will follow UC Student Religious Accommodations for Courses Policy.

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.