21-341: Linear Algebra, Spring 2022

Course web page: Available on CMU Canvas

Lectures: MWF 10:10-11:00am in Porter Hall A18A when permitted, otherwise on Zoom

Professor: Robert Pego    Office: 6127 Wean Hall    Email: rpego AT cmu.edu

• Office hours: 11:00-12:00 WF (live when class is), 8:30-9:30pm M (on Zoom) or just arrange another time.
• Linear Algebra by Charles W. Curtis, Springer. Used mainly for material on abstract fields (chap. 1 sec. 2 pp. 6-16) and determinants (chap. 5).
• Linear Algebra Done Right by S. Axler, Springer, 3rd ed.

Course overview:   Numerous problems of mathematics are understood in terms of linear combinations of various kinds of quantities, some pretty strange. Linear algebra comprises the body of fundamental concepts, properties and transformation rules involved in dealing with these problems. This course provides a mathematically rigorous treatment of linear algebra with coefficients coming from an arbitrary field (of much use in coding theory, for example). Topics studied will include: Fields, abstract vector spaces, subspaces, basis and dimension. Linear transformations and their matrix representation, change of basis, tranformation groups, rank and nullity theorem. Determinants, wedge products. Inner product spaces, eigendata (eigenvalues and eigenvectors), and the singular value decomposition. Multilinear algebra and tensors as time permits.

Learning objectives:   Students should gain and demonstrate understanding of basic definitions and theorems concerning the fundamental topics of linear algebra listed above. They should develop an improved ability and use the concepts, methods and results of linear algebra to compose clear and accurate proofs, and address a variety of problems involving the subject.

Prerequisites:   21-241 (Matrices and Linear Transformations) or 21-242 (Matrix Theory), and 21-373 (Algebraic Structures)

Grading:   Based on 500 points in total:

• 100 points homework
• 200 points midterm exams (one in-class, one take-home, dates TBA)
• 200 points final exam

Homework will be posted on the course Canvas page, and usually will be due Thursdays at 11:59pm on Gradescope. Late homework will not be accepted, but the two lowest homework scores will be dropped. Working homework problems and writing up solutions is essential for your progress. You should work on all the assigned problems (though not all that is turned in can be graded), and more of your choice.

Academic integrity requires that your tests and homework solutions are your independent work and not copied from other sources. The CMU policy is at http://www.cmu.edu/policies/student-and-student-life/academic-integrity.html . On homework you are encouraged to discuss with others and consult other resources to improve your understanding, but it is important to develop your independent capacity to problem-solve. Homework solutions from previous semesters, if available, cannot be used.

Make-up tests are only possible in the case of a documented medical excuse, a university-sanctioned absence (e.g., participation in a varsity sporting event), or a family emergency. Please contact me at the earliest time possible to schedule a make-up.

Health and wellness: Studying mathematics is time-consuming and can be stressful. Make your health a priority, be smart about time management, seek social support, and do ask for help when needed.

Quotes

Mathematics is the art of reducing any problem to linear algebra. ---William Stein

...People can tell you... do it like this. But that ain't the way to learn. You got to do it for yourself. ---Willie Mays