Math 201: Linear Algebra
Spring 2018, MWF 10-10:50 AM or 11-10:50 AM
Krieger 205

Instructor: J. Specter
jspecter at jhu dot edu

Welcome to Linear Algebra! You've found the course website. This is the syllabus. Below you'll find important information about the content, structure and logistics of the course. Please take the time to read it carefully.

What is this course about?

This course is about Linear Algebra.

The text

Linear Algebra with Applications, Fifth Edition, Otto Bretscher. ISBN: 978-0321796943

Your grade

Your grade for this course will be calculated as the weighted average of your grades on the weekly homework assignments (15%, lowest HW grade dropped), weekly quizes (5%, lowest quiz dropped), a pair of midterms (20% each), and a final exam (40%).


There will be three exams in this course. Two in class midterms:

Midterm Exam 1: Monday, March 4,

Practice Midterm (Solutions).

Exam Solutions.

Midterm Exam 2: Monday, April 15

Practice Midterm (Solutions), The TA's Review Problems

Exam Solutions. How was 3(b) Graded?

and a comprehensive final exam:

Final Exam: 9-12 Noon, Thursday, May 9.

All exams will occur in Kreiger 205. Please bring your ID to all exams. Exams must be completed in blue or black pen. The use of textbooks, notes, and calculators will not be permitted.

No make-up exams will be offered in this course. If you have to miss an exam for a documented, legitimate reason, then your final grade will be calculated using your other exam grades.

Graded exams will be returned in section. When you receive a graded exam, please take the time to review the grading and scoring to confirm that no errors have been made. Do this before you leave the classroom. If you take your exam out of the room, it will be assumed that you accept the grade, and under no circumstances will the grade be changed. If you need additional time, please return your exam to your TA and schedule a time to continue your review.

Lecture Notes

Scanned copies of the lecture notes can be found here.

Lecture 1, Solving Systems of Linear Equations (Part I)

Lecture 2, Solving Systems of Linear Equations (Part II)

Lecture 3, The Reduced Row-Echelon Form

Lecture 4, Vectors, Matrices and Linear Transformations

Lecture 5, Examples of Linear Transformations from Geometry

Lecture 6, Dot Products and Orthogonal Projections (Suggested reading: pg. 61-66 in 2.2.)

Lecture 7, Matrix Multiplication (Suggested reading: pg. 75-81 in 2.3.)

Lecture 8, Invertible Matrices (Suggested reading: pg. 88-95 in 2.4)

Lecture 9, Solving Ax = b (Suggested reading: Section 3.1)

Lecture 10, Subspaces of R^n. (Suggested reading: Section 3.2)

Lecture 11, Bases and Dimension (Suggested reading: Section 3.2 p. 128-30; Section 3.3)

Lecture 12, The Rank-Nullity Theorem (Suggested reading: Section 3.3)

Lecture 13, Coordinates (Suggested reading: Section 3.4)

Lecture 14, Similar Matrices (Suggested reading: Section 3.4, pg. 156-159; Section 2.4 on Inverting a 2x2 matrix pg. 93-95)

Lecture 15, Orthogonality and Projection in R^n (Suggested reading: Section 5.1 up to pg. 211)

Lecture 16, Projection and Orthonormal Bases (Suggested reading: Section 5.1 pg. 203-209; Section 5.2 pg. 218-221 (On Gram-Schmidt Process) )

Lecture 18, Orthogonal Transformations (Suggested reading: Section 5.3)

Lecture 18, Approximate Solutions to Ax = b

Lecture 19, Application: Curve Fitting (Suggested reading: Section 5.4, pg. 241-245)

Lecture 20, Volumes of Parallelepipeds in R^n (What is a parallelepiped?)

Lecture 21, Determinants I: Computing the Determinant (Suggested reading: Section 6.2)

Lecture 22, Determinants II (Warning! There is a subscripting error in these notes on page 19. The subscripts should {i,1} and not {1,i}). Suggested reading: Section 6.3)

Lecture 23, Diagonalization (Suggested reading: Section 7.1)

Lecture 24, Finding the Eigenvalues of a Matrix (Suggested reading: Section 7.2)

Lecture 25, Finding the Eigenvectors of a Matrix (Suggested reading: Section 7.3)

Lecture 26, Matrix Powers and Dynamical Systems (Suggested reading: Section 7.1 pg. 316-323; See Section 7.4 for additional examples)

Lecture 27, Complex Eigenvalues (and Eigenvectors) (Suggested reading: Section 7.5 (review complex numbers: pages 363-367))

Lecture 28, Matrix Powers with Complex Eigenvalues (Suggested reading: Section 7.6; λ = x + iy, a non-real number such that |λ| > 1.)

Lecture 29, Matrix Powers with Complex Eigenvalues II (See Theorem 7.5.3 and Example 3 of 7.6)

Lecture 30, Symmetric Matrices (Suggested reading: Section 8.1)

Lecture 31, Quadratic Forms I (Suggested reading: Section 8.2. This lecture will focus on quadratic forms in 2 variables.)

Lecture 32, Quadratic Forms II (Suggested reading: Section 8.2)

Lecture 33, Singular Value Decomposition (Suggested reading: Section 8.3)


Homework accounts for 15% of the grade for this course. It will be assigned weekly, and your homework grade will be the average of your weekly assignments. Homework is due at the beginning of class on its posted due date. Most weeks, assignments will be posted on Friday and will be due on the following Friday. Homework must be written legibly and stapled when necessary. No late homework will be accepted.

You are encouraged to talk to your classmates about the material covered in class and collaborate on homework. However any assignment you pass in must be primarily your own work. To avoid the pitfalls of plagiarism, please write up your assignments alone and independently. If you've worked on a problem with another student, please acknowledge that collaboration in your write up (of that problem).

Homework assignments will appear weekly - posted to the following table.

Week 1 Please read sections 1.1,1.2, and 1.3
Week 2 1.1: 12,14,16,43;
1.2: 6, 10, 12;
1.2: 30,48, 50;
1.3: 1, 22.
Week 3 1.3: 16,18,19,35,36;
2.1: 6, 24, 26,42;
2.2: 2, 6, 7, 16, 32, 36.
Week 4 2.3: 7, 8 (you may alternatively do 2.1: problem 13), 29, 30, 38;
2.4: 1, 6, 8, 14, 28;
3.1: 6, 15, 24, 34 (Hint: use projection), 42.
Week 5 3.1: 20,21,41;
3.2: 5,6,7,30,42,46;
3.2: 14,20,26;
3.3: 24,25,28,39.
Quiz 4 Solutions
Week 6 3.4: 4, 6, 16, 18, 26, 28, 30, 38, 39, 56, 60, 67. Solutions
Quiz 5
Week 7 5.1: 4,6,11,26,28.
5.1: 40,41,42,43,44,45,46.
5.2: 2,6,14,29,34.
Week 8 5.3: 2,4,36,40,43,67,69,70.
5.4: 2,9,22,24,26,30,32,36.
Week 9 6.2: 2, 4, 6, 8,40, 44, 46, 66
6.3: 1, 2, 3, 4 , 7, 14, 29, 47.
Week 10 7.1: 15, 16, 18, 56, 62;
7.2: 2, 10, 12, 15, 16, 32;
7.3: 4, 6, 14, 18, 20, 52;
Bonus: 7.3, 54.
Quiz 8 Solutions
Week 11 7.1: 72;
7.4: 4, 6, 10, 12, 26, 34, 38;
7.5: 9, 20, 22, 24, 30;
7.5: 33, 34; (these two problems will be among those graded)
Due MONDAY, 4/22
Week 12 7.5: 14, 16, 17, 48;
7.6: 32, 41;
8.1: 6, 8, 22, 26, 42;
8.2: 3, 6, 18, 20, 27, 38;
Due MONDAY, 4/29

Remember to show both your work and your reasoning on your homework solutions. The DUS Rich Brown has written a wonderful note on homework presentation which you can find here.

Sections, Quizzes, and Office Hours

The goal of the section meetings is to help you bring the theory presented in the lectures into practice. Please come with questions! Graded homework will be passed back during section meetings so it is important that you attend only the section assigned to you.

  Meeting Times Location Instructor Email: < >
Section 1 Tuesdays 3:00-3:50 Shaffer 100 Benjamin Dees bdees1
Section 2 Tuesdays 4:30-5:20 Hodson 211 Patrick Kennedy pkenne16
Section 3 Thursdays 1:30- 2:20 Hodson 316 Patrick Kennedy pkenne16
Section 5 Tuesdays 1:30-2:20 Hodson 316 Jeff Marino jmarino9
Section 6 Tuesdays 3-3:50 Maryland 309 Jeff Marino jmarino9
Section 7 Thursdays 3:00-3:50 Hodson 203 Xiangze Zeng xzeng12
Section 8 Thursdays 4:30- 5:20 Hodson 211 Xiangze Zeng xzeng12

During each section meeting you'll be given a short quiz. These quizzes account for 5% of your grade.

In addition to the sections each of the TA's and myself will hold weekly office hour.

My office hours are:
J. Specter: Wednesdays 12:15 - 1:45 in Krieger 419 or by appointment.

TA office hours:

Marino: Thursdays 12-1 in Krieger 200. Help room Wednesday 11-1.

Kennedy: Wednesdays 5-7 in Krieger 207.

Dees: Thursdays 3-4 PM in Krieger 201. Help room hours are 11-1 on Mondays

Zeng: Thursdays 11am-1pm in Krieger 213.

Additional resources

Students with disabilities

Students with documented disabilities or other special needs who require accommodation must register with Student Disability Services. After that, remind the instructor of the specific needs at least two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.

There may be a student in this class who requires the services of a note taker. This is an opportunity to share notes through the Student Disability Services Office. If you are interested in performing this service, please register as a notetaker with Student Disability Services via the following URL:

JHU ethics statement

"Undergraduate students enrolled in the Krieger School of Arts and Sciences or the Whiting School of Engineering at the Johns Hopkins University assume a duty to conduct themselves in a manner appropriate to the University's mission as an institution of higher learning. Students are obliged to refrain from acts which they know, or under circumstances have reason to know, violate the academic integrity of the University. [The JHU Code of Ethics]"

Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. If a student is found responsible through the Office of Student Conduct for academic dishonesty on a graded item in t