Math 231

(Last updated Monday, December 12 @ 2pm)

Course Information

  • Instructor: Professor Nicholas Vlamis

  • Instructor Office: 507 Kiely

  • Instructor Email:

  • Office Hour: Monday 2–3pm (or by appointment)

  • Syllabus

  • Textbook: Anton, Howard and Kaul, Anton. Elementary Linear Algebra, twelfth edition. John Wiley & Sons, 2019.

Week 16 (Week of December 12)

Discussed least-squares solutions to linear systems.
  • Please take a couple of minutes to fill out the course evalution:

  • Final Exam Information:

    • Date/Time/Location: Monday, December 18 11am-1pm in Kiely 422 (our usual classroom).

    • Office Hours: Thursday 2-3pm and Friday 11am-12pm (in my office: 507 Kiely)

    • Information Sheet for Final Exam

Week 15 (Week of December 5)

  • Quiz 11 (our last quiz) will be given in the beginning of class on Wednesday, Dcember 7.

  • Monday's Class Notes

Went over the project theorem, the Gram–Schmidt process, and several examples. Proved the QR-decomposition theorem and discusses how this is used to compute eigenvalues.
  • Note: Final Exam is on Monday, December 18 11am-1pm.

Week 14 (Week of November 28)

Introduced the notions of an inner product, inner product space, norm, distance, angle, and orthogonality. Went over some examples, including an inner product on the space of continuous function on a closed interval. Discussed orthogonal complements and introduced the notion of an orthonormal basis.
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Week 13 (Week of November 21)

Introduced the notion of diagonalizable matrices and how to tell if a matrix is diagonalizable.

Week 12 (Week of November 14)

Went over example of finding eigen values and their associated eigenspaces. Introduced the notion of similar matrices and discussed the properties they share.

Week 11 (Week of November 7)

Fixed a theorem statement from last time. We explored the derivative as a linear transformation and computed it's matrix when restricted to polynomials of degree at most 2. Spent half the class talking about homework. Introduced the notions of eigenvalue, eigenvector, and eigenspace. Investigated how to find these objects for a given matrix.
  • No homework this week. Study for the exam! (A homework on eigenvectors will be posted next week.)

  • Solutions to Quiz 9

Examples of eigenfaces.

Week 10 (Week of October 31)

Proved several theorems regarding linear independence and spanning, which allowed us to introduce the notion of coordinates and dimension.
  • Note: Exam 2 is on Wednesday, November 16. Exam 2 will cover Sections 2.1-2.3, 3.1, 3.2, 4.1-4.6, 4.8, and 4.9 from the textbook.

  • Wednesday's Class Notes

Introduced the notions of row space, column space, null space, rank, nullity, and orthogonal complements.

Week 9 (Week of October 24)

Further discussed examples of subspaces. Introduced spanning sets. Introduced the notions of linear independence and a basis of a vector space. We proved severeal theorms dealing with checking if a set is linearly independent or not.

Week 8 (Week of October 17)

Introduced dot product, norm, and distance. Proved the Cauch-Schwarz Inequality and the Triangle Inequality. Gave an application of the dot product to comparing two documents. See Yitang Zhang give a talk intended for undergraduates on number theory. Prof. Zhang made a huge discovery a few years ago towards the Twin Prime Conjecture. Pizza will be served. See above link for details. Introduced vector spaces and subspaces and went over many exmaples.

Week 7 (Week of October 10)

Established several properties of the determinant and went over Cramer's rule.

Week 6 (Week of October 3)

  • Exam 1 was held on Monday, October 3.

  • No class Wednesday, October 5 or Monday, October 10 (this is CUNY-wide)

  • Our next class is on Wednesday, October 12, and we will have a quiz based on Homework #5.

  • Solutions for Exam 1

These solutions were written on a draft of the exam, so you may see minor differences in the wording of some questions, but the content is the same.

Week 5 (Week of September 26)

  • Quiz 4 will be given in the beginning of class on Wednesday, September 28

Discussed the composition of matrix transformations and used these ideas to prove classical trig identites (e.g. angle-sum identities and double-angle identities). We then introduced the definition of the determinant and went over examples computing the determinant.Discussed the relationship between determinant and area. Established basic properties of determinants of elementary matrices and gave an algorithm for computing determinants using Gauss Elimination.
  • Homework #5 (Associated quiz will be on Wednesday, October 12)

A proof-by-picture of the fact that the determinant of a 2x2 matrix is the absolute value of the area of the parallelogram determined by its row vectors.

Week 4 (Week of September 19)

Introduced the notion of diagonal, triangular, and symmetric matrices. Begin our study of matrices as transformations.
  • Note: We have class on Thursday, September 29 (CUNY will follow a Monday schedule that day)

  • Note: Exam 1 is in two weeks on Monday, Oct 3. It will cover Sections 1–9 of Chapter 1.

  • Wednesday's Class Notes

Introduced the notion of a linear transformation from R^n to R^m and showed a one-to-one correspondence between linear transformations and matrix transformations in this setting. We also went over some standard examples of linear transformations from geometry (reflections, projections, rotations), and computed their associated standard matrix.

Week 3 (Week of September 12)

Discussed basic properties of matrix algebra and introduced the inverse of a matrix. Showed how to realize elementary operations as matrix multiplication by elementary matrices, proved a theorem giving several equivalent conditions for a matrix to be invertible, gave an algorithm for finding the inverse of a matrix, and proved that every linear system has 0, 1, or infinitely many solutions.

Week 2 (Week of September 5)

  • No class Monday, September 5 (Labor Day)

  • Quiz 1 will be given in the beginning of class on Wednesday, September 7

  • Syllabus update: your lowest quiz grade will be dropped.

  • Wednesday's Class Notes

Finished discussing using Gauss–Jordan Elmination to solve systems of linear equations, introduced a slate of definitions and operations involving matrices.

Week 1 (Week of August 29)

  • Reading: Read Sections 1.1 and 1.2 in the course textbook. Make sure to read the definition of pivot position, pivot row, and pivot column on page 22.

  • Monday's Class Notes

Introduced a simplified version of Google's PageRank Algorithm, and introduced the notion of a linear system.
  • For those interested, the Google example was lifted from Strogatz's book The Joy of x, which has nice examples of math in the real world.

  • Contact Information Form: Please fill out this form so that I have your preferred email address. I will use the address you provide to send out class announcements.

  • Homework #1 (Associated quiz will be on Wednesday, September 7) ***Updated Wed. Aug. 31 @ 12:30pm

  • Wednesday's Class Notes

Introduced the augmented matrix associated to a linear system, the notion of reduced row echelo form, and learned the Gauss-Jordan Elimination algorithm for finding the reduced row echelon form of a matrix.