Math 201: Linear Algebra
Fall 2017, MWF 10-10:50 AM
Bloomberg 272

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 13 weekly homework assignments (15%, lowest HW grade dropped), a pair of midterms (23% each), and a final exam (39%).


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

Midterm Exam 1: 10-10:50 a.m. Monday, October 9,
Midterm Exam 2: 10-10:50 a.m. Monday, November 13

and a comprehensive final exam:

Final Exam: 9-12 Noon, Wednesday, December 13.

All exams will occur in Bloomberg 272. 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

I will post notes from each of the lectures (after lecture 5) online on our instagram. Yes, we have an instagram.


Homework accounts for 15% of the grade for this course. It will be assigned weekly, and your homework grade will be the average of the 13 highest grades from 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 0 Please read sections 1.1,1.2, and 1.3
HW 1 In section 1.1, please do exercises 6,12,13,14,21. For bonus, do 1.1 exercise 30. In section 1.2, please do exercises 1,4,10,18,25. Bonus: 31 Problems Mock HW Solution due 9/8
HW 2 Read section 2.1 and 2.3 for Monday, 2.2 for Wednesday. Do exercises 17,20,36,47,57 in Section 1.3; Do exercises 6,14,24,25,26,27,28,29,30,36,63,64 in Section 2.1; Do exercises 2,4,6,7,56,64 in section 2.3. No Bonus this week. Problems due 9/15
HW 3 Today's lecture and next Monday's lecture could be augmented by reading 5.3 (this is optional and you should beware that there are terms used that we have not defined in class - consult your textbook). For the complex numbers: read 363-366 in our textbook.

The relevant reading for next Monday is 3.2 (however before reading this section please look at definition 3.1.5 and 3.1.3). Please read 3.1 for Wednesday (optionally for Monday). Please read 3.3 for Friday.

The homework for this week is: 2.2 problems 8,9,19,20,23,24,32,36, bonus: 47; 5.3 problems 33,34,35,36 Bonus: 64; 3.1 problems 4,6,8,37; 3.2 problems 6, 12, 19, 32, 42, 53
Problems + Lecture notes
due 9/22
HW 4 Please read chapter 2.4 for Monday. Of particular importance are pages 89-93 (up to example 3). The lecture will relate to these pages. Compare Example 1 to our calculation in Lecture 10.

For Wednesday, please read the beginning of chapter 3.4: pages 147-153. Definition 3.4.1 will be important, as will theorem 3.4.3. Work through examples 1 and 2. It might also be of interest to look at examples 9 and 10 (at the end of the chapter). These examples pertain to finding the best bases (coordinates) to represent reflections in the plane.
There's no reading for Friday. Work on your HW, have fun, and be safe.

The homework for this week is: problem 54 in 3.2; problems 21,22,27,62,64,68 in 3.3; problems 12,14,56,76 in 3.4; problems 1,4,14,52,56 in 2.4; problems 1,7 in 7.5; Bonus 90,108 in 2.4. Problems + Lecture notes
due 10/2
Practice Mideterm Practice Midterm Solutions Page4

Disclaimer: the following notes are from the review session. They should only serve as a reminder of what was said there. I do not claim that they are comprehensive or (even) comprehensible. Notes from Review Session

Solution Guide for Midterm 1
HW 5 Next week, I will continue our discussion of coordinates. Please finish reading chapter 3.4 for Wednesday.

Concerning the test: there will be a review session Wednesday night (time and location TBA). Notes from the review session will be posted on instagram on Thursday.

The goal for next week is to study the problem of deciding if a pair of 2x2 matricies are similar. I will define this notion on Monday. Periodically throughout the week (but after Monday's lecture), open the book to pages 368-370 and look at example 6,7, and 8. Ask yourself "do I understand this yet?" If the answer is NO, close the book without reading, and study for the test. If the answer is YES, do the same.

Please do problems: 38 in 2.2; 74 in 3.3; 57,59,60,61,63,68,70 in 3.4; 13,14,15,16 in 7.5; No bonus this week.
due 10/6
HW 6 No reading this week.

Please do:

problems 28,29,30,79,81,82 in section 3.4; problem 39 in section 7.2; problems 55,58,59 in section 7.4; do problems 1-4 here: problems. Problem 5 is bonus.
due 10/13
HW 7 For Monday, please read pages 310-315 of section 7.1 up to Example 6, and look at figure 10 on page 323 and read the paragraph below. Goal: think about how diagonal matricies generalize the example from class. What is the minimal polynomial of a diagonal matrix?

On Monday, (and perhaps Wednesday depending on timing) after a brief aside on diagonal matrices, I will talk about shears and their higher dimensional analogs. Please reread the relavent material in section 2.2 on pages 68-70.

On the HW, use only methods introduced in class. Other methods (for example using characteristic polynomials or complex diagonialization) will receive partial or no credit. Please do exercises 15,16,17 in section 7.1, exercise 45 in section 7.2, exercises 1,2,4,6,21,24 (in these last two exercises E_* is our E_*,A) in section 7.3, exercises 2,4,5,6,14,15 in 7.4, exercises 20,21,22 in section 7.5; Bonus 53 in section 7.3
due 10/23
HW 8 Please read these notes On how to compute eigenvectors of a matrix by projecting onto eigenspaces . Afterwards take the matrix in problem 4 part (3) of last week's HW (in the packet), and try to diagonalize it by first finding its eigenvalues (following lecture 20), and then finding a basis of eigenvectors (using the strategy described these notes). I will discuss these notes on Monday. Please read this note On why the minimal polynomial of a diagonal matrix has no repeated roots.

There are some new terms on this week's HW:

(1) Isomorphism: A fancy word for invertible linear transformation.
(2) Linear Space: You can think about nxn matricies as R^(n^2).
(3) Definition: A matrix is called symmetric if A = A^T. (in terms of matrix entries this means a_{i,j} = a_{j,i}.)

HW: Problems 65,66,68 in 7.1; Problems 13,15,16,33,34,37,48,49,50, 54(a) in 7.3; 11,28,61 in 7.4.
due 10/27
HW 9 The relevent section from the book for today's lecture are pages 279-284, 294-300. Read section 6.1 at your own risk.

On Wednesday, I will talk about a few more ways to calculate the determinant and define characteristic polynomials. Please read 284-288 in section 6.2 and section 7.2.

Please do: 17,20 in section 6.1; 3,4,9,10,31,32,49 in section 6.2; 2,8,10,11,29,30 in section 7.2; 52,54(b,c,d) 7.3; 32,33 in 7.4; 41 in 7.6;
due 11/8
Practice Midterm 2 I've been asked to post a practice midterm. Personally, I think it's rather early (it's only been 3 weeks since the last midterm!) However, for those of you wondering what the midterm will look like here's a good approximation: Practice Midterm Solutions (here's an alternate version of this first practice test: Practice exam with a modified version of question 3.)

Here's a word problem: it's a harder version of the third question on the second midterm. Try it out: NetworkProblem

PSA: Diagonalization for 2x2 matricies should not require you to do any heavy linear algebra. If you find yourself multiplying matrices (or vectors) or using row reduction, you are NOT solving the problem in the most efficient way possible. We have formulas for the minimal polynomial, the roots of a quadratic polynomial, eigenvectors (projection formula), and the inverse of a matrix - use these formulas, they will save you time on the midterm! Here are some examples: Diagonalization Examples (2x2)

Here's a second Practice Midterm. Solutions
Errata:Rank 0 on page 1 should be rank 1; eigenvalue 0 on page 1 should be eigenvalue 1. (2), problem 2a there is a mistake in the expansion of the characteristic polynomial. On the 6th line, - (2 - λ)^3 - 2 (2 - λ) is correct, but its expansion is not. The characteristic polynomial is -λ^3 + 6λ^2 - 10λ + 4.

Here's a third Practice Midterm. Solutions
A review session happened on Monday night.

Midterm 2 Solutions

There will be no HW 10.
HW 11 Section 5.1: 9,26,28,29,32,37; Section 5.2: 3,4,7,11,19,28,32,35; Section 5.3: 35; Section 5.4: 8,10,21,31,32,35,38,39. due 11/27
HW 12 Last week we saw that to solve Ax = b, approximately, one solves A^TAx = A^Tb. This week we will discuss symmetric matrices (of which A^TA is an example). These are the matrices which appear most frequently in applications of linear algebra. Please read section 8.1 for monday, 8.2 for wednesday, and 8.3 for friday.

HW: Section 8.1 Problems: 1,2,4,5,8,10,11,16,19,21,22,24,25,26,27,28,31,38,39,40,41,42,43; Problem 5 on the practice test. 5.3 problems: 43,69.
due 12/1
HW 13 Section 8.2, problems (1,4,15),17,18,19,22,23,26; Section 8.3, problems 4,5,6,17,18,27,30. due 12/8
Practice Final Here's the first practice final. Selected solutions: Problem 1 Problem 2 Problem 3 Problem 5 Here's the second practice final. Solutions: Problem 1 Problem2 Problem 3Problem 4 Problem 5 Here's a problem of type #1 Restricting allowed moves in Gaussian Elimination Notes from the review session Solution

Sections 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: <>
Lecture 1 Tuesdays 1:30-2:20 Hodson 313 Tianyi Ren tren2
Lecture 2 Tuesdays 3-3:50 Maryland 309 Alec Farid afarid1
Lecture 3 Tuesdays 4:30- 5:20 Maryland 202 Alec Farid afarid1
Lecture 4 Thursdays 1:30-2:20 Shaffer 300 Tianyi Ren tren2
Lecture 5 Thursdays 3-3:50 Hodson 305 Lance Corbett lcorbet5
Lecture 6 Thursdays 4:30- 5:20 Hodson 210 Lance Corbett lcorbet5

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

My office hours are:
J. Specter: Wednesdays 11:15 - 1:15 in Krieger 419 or by appointment.

TA office hours:

Lance Corbett: Mondays in Krieger 207 from 3-4 (or by appointment),
Alec Farid: Thursdays in Krieger 207 from 9-10 a.m, and
Tianyi Ren: Wednesdays in Krieger 201 from 3:30-5:30.

The TA does not have to be the TA of your section to attend their office hour.

In addition to holding office hours each of our TAs work in the math help room (Krieger 213). Lance can be found there on Tuesday nights from 7-9, Tianyi's hours are Thurdays from 5 to 7, and Alec works from 9-11 on Friday mornings.

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 this course, the student will receive a score of zero for that assignment, and the final grade for the course will be further reduced by one letter grade.