Robert B. Kusner

Director of the Center

(This page is like a cathedral: always under reconstruction ;-)

Math 235.06 Spring 2021: Introduction to Linear Algebra

• • • Much of this is mirrored on Rob Kusner's Moodle page, but (for your convenience) here's how to get the e-text and sign up for on-line homework for Rob's section (235.06) only!

Accommodation Policy Statement:

Rob is committed to providing an equal educational opportunity for all his students. A student with a documented physical, psychological, or learning disability on file with UMass Amherst Disability Services (DS) may be eligible for academic accommodations to help them succeed in this course. If you have a documented disability that requires an accommodation (in addition to anything which you expect was sent to Rob by DS) please contact Rob DIRECTLY at during the FIRST TWO WEEKS of the semester so that we can make appropriate arrangements. (Information on services and materials for registering with DS are available on the DS website.)

Course-wide Videos:

We are making a series of our own short videos to help you learn linear algebra this semester; please watch and think about them before each class!

Our Learning Sessions:

M & W afternoons at 4 (or other later times to be announced). All our sessions are via Zoom (Rob will email a link before the session).

Office Hours:

Immediately after each class, and (if we don't already have a student-group-hosted meeting the same day) almost any evening on Zoom with an advance appointment via email:

If you are Waiting to Enroll:

Please contact Rob by email at to introduce yourself, and in a short time we should be able to officially enroll you (we don't want to deal with over-rides until we know they're necessary).


Familiarity with basic algebra, vector geometry and (later in the course) a bit of calculus and differential equations.

Common Guidelines & Weekly Topics Schedule:

Math 235 is an introductory course on linear algebra, covering systems of linear equations, matrices, linear maps, determinants, vector spaces, eigenvalues and eigenvectors, and orthogonality.

All learning sections of Spring 2021 Math 235 will try to follow the Common Guidelines & Weekly Topics Schedule detailed on the Syllabus at my Moodle page, but here's a handy synopsis of our weekly schedule (this a guideline which may be modified as necessary):

[Week 1] 2/1: 1.1 Systems of linear equations; 1.2 Row reduction and echelon forms; 1.3 Vector equations.

[Week 2] 2/8: 1.4 The matrix equation AX=B; 1.5 Solution sets of linear systems.

[Week 3] 2/15: 1.7 Linear independence; 1.8 Introduction to linear maps.

[Week 4] 2/22: 1.9 The matrix of a linear map; 2.1 Matrix operations.

[Week 5] 3/1: 2.2 The inverse of a matrix; 2.3 Characterizations of invertible matrices.

Common Midterm I: 3/4 on GradeScope 7-9PM EST

The topics for the first midterm may include the following sections of the textbook: 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.8, 1.9, 2.1, 2.2, 2.3
Please work through the problems at the end of Chapters 1 & 2 before the exam.
Here are some Midterm I review suggestions (courtesy of an inspiring former instructor, Pat Dragon).
Check your email from late February for the practice exam for Midterm I!
For your convenience, here are the course chair's Midterm I practice problems/notes/solutions.

[Week 6] 3/8: 3.1 Introduction to determinants; 3.2 Properties of determinants.

[Week 7] 3/15: 3.2 (continued); 3.3 Cramer's rule, volume and linear maps.

[Week 8] 3/22: 4.1 Vector spaces and subspaces; 4.2 Null space (kernel), column space (image) and linear maps.

[Week 9] 3/29: 4.3 Linearly independent sets and bases; 4.4 Coordinate systems.

[Week 10] 4/5: 4.5 The dimension of a vector space; 4.6 The rank+nullity theorem.

Common Midterm II: 4/8 on GradeScope 7-9PM EDT

The topics for the second midterm may include the following sections of the textbook: 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6
Please work through the problems at the end of Chapters 3 & 4 before the exam.
Here are some Midterm II review suggestions (once again, courtesy of Pat Dragon).
Check your email from early April (no foolin'! ;-) for the practice exam for Midterm II!

[Week 11] 4/12: 5.1 Eigenvectors, eigenvalues and eigenspaces; 5.2 The characteristic equation.

[Week 12] 4/19: 5.3 Diagonalization; 5.5 Complex eigenvalues.

[Week 13] 4/26: 6.1 Inner product, length and orthogonality; 6.2 Orthogonal sets; 6.3 Orthogonal projection.

[Week 14] 5/3: 6.4 The Gram-Schmidt process; 6.5 Least squares solutions (if time permits).

Final Exam: 5/?*

The topics for the final exam may include the following sections of the textbook: 4.5, 4.6, 5.1, 5.2, 5.3, 6.1, 6.2 (and possibly some of 6.3, 6.4, 6.5).
Please work through the problems at the end of Chapters 5 & 6 before the exam.
Here is a Final review (revue?!) at long last (and again, of course, courtesy of Pat Dragon).

*All exam dates above are tentative!!!

UMass Amherst's Academic Honesty Policy

Since the integrity of the academic enterprise of any institution of higher education requires honesty in scholarship and research, academic honesty is required of all students at the University of Massachusetts Amherst. Academic dishonesty is prohibited in all programs of the University. Academic dishonesty includes but is not limited to: cheating, fabrication, plagiarism, and facilitating dishonesty. Appropriate sanctions may be imposed on any student who has committed an act of academic dishonesty. Instructors should take reasonable steps to address academic misconduct. Any person who has reason to believe that a student has committed academic dishonesty should bring such information to the attention of the appropriate course instructor as soon as possible. Instances of academic dishonesty not related to a specific course should be brought to the attention of the appropriate department Head or Chair. Since students are expected to be familiar with this policy and the commonly accepted standards of academic integrity, ignorance of such standards is not normally sufficient evidence of lack of intent (please visit the UMass Dean of Students Office for any updates to the official policy).


We're planning more ways for students to participate individually in the course! Please stay tuned!!

In addition to what you'll find on the Coursewide Spring'21 Common Guidelines... or the Fall'20 Common Guidelines... pages, here's a bit more possibly useful (or useless ;-) information:

⇒ Several past exams are available here or at the Kusner's Math Classes page.

⇒ Before the first midterm, you may enjoy reviewing Basic Linear Algebra in 7 Easy Pages prepared by my former student (2011-14) Andrew Maurer (recently in grad school at the University of Georgia; his PhD advisor Dan Nakano took my linear algebra course at Berkeley in the early 1980's - years later Dan told me that my course made him want to become a mathematician).

⇒ Before the second midterm, you may appreciate this Linear Algebra Review Sheet by a semi-recent (Spring 2017) student Jonah Palmer (with a couple comments from yours truly).

⇒ For the hardcore linear algebra students: the usual multiplication algorithm has complexity O(n^3) for n × n matrices; here's a recent paper on the latest complexity bound: O(n^2.3726...). (Conjecture: O(n^2) is best.)

⇒ The last part of the course uses the relation AE=ED (for any matrix E whose columns form a basis of real eigenvectors of a real 2 × 2 matrix A with distinct real eigenvalues) to factor A=EDE^{-1}, where D is the real diagonal 2 × 2 matrix whose nonzero entries are the eigenvalues, i.e. A is similar to D. This is useful to compute powers of A and the exponential of A. But what if A has repeated real eigenvalue (a, a), or if the eigenvalues are complex conjugate pairs (a+bi, a-bi) with nonzero b? How does one decompose A=aI+bJ where J^2=-I in the latter case, or decompose A=aI+N where N^2=0 in the former? Some texts (e.g. Bretscher) find a 2 × 2 matrix aI+bJ_o to which A is similar; here J_o is the standard 90-degree rotation matrix. Here's another approach, suggested by my emeritus colleague Arunas Rudvalis, which seems simpler - and more general since it also deals with the N (nilpotent) case.

!!!DRAFT!!! (Still under reconstruction! ;-)