Director of the Center
(This page is like a cathedral: always under reconstruction ;-)
• • • 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 profkusner@gmail.com
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:
profkusner@gmail.com
If you are Waiting to Enroll:
Please contact Rob by email at profkusner@gmail.com 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).
Prerequisites:
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
[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
[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).
*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).
Remarks:
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.
Math 235.06 Spring 2021: Introduction to Linear Algebra
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.
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!
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).
!!!DRAFT!!! (Still under reconstruction! ;-)