# Math 231

(Last updated Wednesday, May 17 @ 10:45am)## Course Information

Instructor: Professor Nicholas Vlamis

Instructor Office: 507 Kiely

Instructor Email: nicholas.vlamis@qc.cuny.edu

Class Meeting: Monday/Wednesday 10:05–11:55am in Kiely 242

Office Hour: Monday/Wednesday 4–5pm (or by appointment)

Textbook: Lay, David. Linear Algebra and Its Applications, Sixth Edition. Pearson, 2021.

Optional textbook: Interactive Linear Algebra by Dan Margalit and Joseph Rabinoff.

## Week 16 (Week of May 15)

Final Exam on Monday, May 22 from 11am–1pm in Kiely 242 (our normal classroom). The exam is cumulative. The exam will cover the following sections from the textbook: Chapter 1 (except for Sections 1.6 and 1.10), Sections 2.1, 2.2, 2.3, 2.8, 2.9, 3.1, 3.2, 5.1, 5.2, 5.3, 5.9, 6.1, 6.2, 6.3, 10.1, 10.2

Office Hours:

Friday 10:30–11:30am in my office, Kiely 507

Sunday 4–5pm on Zoom (https://us02web.zoom.us/j/85682379027?pwd=WVRUbG9BSDltNWxEWVpzSGhCVlB0UT09)

## Week 15 (Week of May 8)

Helpful reading: Sections 6.4 and 6.5

Homework Assignment #12 (Not collected, but relevant to final exam.)

Monday's Class: Introduced the notion of a least-squares solution to a linear system, proved such a solution exists, characterized it was unique, and gave a formula for the solution in terms of the QR decomposition theorem. Explained how linear regression is a least-squares problem.

Wednesday's Class: Introduced vector spaces and generalized the notions of linear transformation, linear combination, spanning, linear independence, basis, dimension, and coordinates to this setting through many examples. Defined an inner product and inner product space focusing on the example of Fourier approximations.

Course evaluations: http://www.qc.cuny.edu/evaluate (Please fill them out!)

Final Exam on Monday, May 22 from 11am–1pm in Kiely 242 (our normal classroom). The exam is cumulative. The exam will cover the following sections from the textbook: Chapter 1 (except for Sections 1.6 and 1.10), Sections 2.1, 2.2, 2.3, 2.8, 2.9, 3.1, 3.2, 5.1, 5.2, 5.3, 5.9, 6.1, 6.2, 6.3, 10.1, 10.2

## Week 14 (Week of May 1)

Helpful reading: Sections 6.2 and 6.3

Homework Assignment #11 (Due Wednesday, May 10)

Monday's Class: Proved the orthogonal projection theorem and introduced the Gram–Schmidt Process for finding an orthonormal basis for a subspace.

Wednesday's Class: Went over a detailed example going through the Gram–Schmidt Process. Proved the QR Decomposition theorem and gave an application to computing eigenvalues.

## Week 13 (Week of April 24)

Helpful reading: Sections 6.1 and 6.2

***Homework Assignment 9 will be collected on Monday, May 1 and graded as a quiz.

Homework Assignment #10 (Due Wednesday, May 3)

Monday's Class: Introduced distance, angles, and orthogonality. Discussed how to compare documents using the dot product. Introduced the orthogonal complement.

Wednesday's Class: class was cancelled due to me being sick. To make up the class, I've recorded a lecture in three parts. Please watch by Monday, May 1. (Here is a pdf copy of the notes.)

## Week 12 (Week of April 17)

Helpful reading: Sections 5.9, 10.1, 10.2

Wednesday's class will end early at 11:30am.

Homework Assignment #9 (Due Wednesday, April 26)

Monday's Class: Introduced (strongly connected) directed graphs, (regular) Markov chains, and stochastic matrices. We proved that a regular stochastic matrix has a unique stochastic eigenvector whose eigenvalue is 1.

Wednesday's Class: Defined Google PageRank using the theory of Markov Chains. (Re-)Introduced the dot product and defined the associated norm.

## Week 11 (Week of April 3)

Helpful reading: Sections 5.1, 5.2, and 5.3

Homework Assignment #8 (Due Wednesday, April 19)

Exam 2 was on Monday, April 3.

CUNY Spring Recess: Wednesday, April 5 through Thursday, April 13

## Week 10 (Week of March 27)

Helpful reading: Sections 5.1, 5.2, and 5.3

***Exam 2 is on Monday, April 3***. The exam will cover the content of Sections 2.1, 2.2, 2.3, 2.8, 2.9, 3.1, and 3.2. You may bring a sheet of notes. The statement of the invertible matrix theorem will be given on the test (you can see what it looks like by clicking the following link: Invertible Matrix Theorem).

Homework Assignment #8 will be posted by Monday (and will be due Wednesday, April 19).

Monday's Class: Introduced eigenvectors, eigenvalues, and eigenspaces.

Wednesday's Class: Introduced diagonalization and proved a theorem relating diagonalizability and the existence of an eigenbasis.

## Week 9 (Week of March 20)

Helpful reading: Sections 2.9, 3.1, 3.2

Exam 2 is on Monday, April 3. The exam will cover the content of Sections 2.1, 2.2, 2.3, 2.8, 2.9, 3.1, and 3.2.

Homework Assignment #7 (Due Wednesday, March 29)

Monday's Class: Introduced coordinate systems for subspaces. Defined the determinant and went over examples of how to compute it.

Wednesday's Class: Went over various properties of the determinant.

## Week 8 (Week of March 13)

Helpful reading: Sections 2.8 and 2.9

Homework Assignment #6 (Due Wednesday, March 22)

Monday's Class: Found basis for column space of a matrix. Proved that any two bases for a subspace have the same number of vectors. Defined dimension.

Wednesday's Class: Introduced rank and nullity. Proved the rank theorem and discussed examples. Extended inverse matrix theorem. Motivated and stated basis theorem.

## Week 7 (Week of March 6)

Helpful reading: Sections 2.2, 2.3, and 2.8

Homework Assignment #5 (Due Wednesday, March 15)

Monday's Class: Introduced the notion of an invertible matrix. Discussed several characterizations of invertibility.

Wednesday's Class: Proved that one-sided inverse of a matrix is the inverse. Introduced the notions of subspace, column space of a matrix, row space of a matrix, null space of a matrix, and basis. Went over several examples, including establishing that the span of a collection of vectors is a subspace. Proved that the null space of a matrix is a subspace. Through an example, discussed the algorithm for finding a basis for the null space of a matrix.

## Week 6 (Week of February 27)

Helpful reading: Sections 2.1

Exam 1 on Wednesday

Exam covers Chapter 1 of text book (except for Sections 1.6 and 1.10).

You may bring a single sheet of notes (8.5"x11" paper) to the exam. You may write theorem statements and definitions and brief notes to yourself, but you may not include solutions to problems or proofs of theorems.

Monday's Class: Introduced several matrix operations. The majority of class was spent defining matrix multiplication.

## Week 5 (Week of February 20)

Helpful reading: Sections 1.8 and 1.9

Required Reading: Section 1.9 Subsection labelled Existence and Uniqueness Questions (page 77–81).

We did not have class on Monday (CUNY closed), but we had class on Tuesday.

Tuesday's Class: Introduce the notion of linear transformations and matrix transformations, and proved that every linear transformation from R^n to R^m is a matrix transformation.

No lecture on Wednesday, February 22.

Exam 1 is next week on Wednesday, March 1. It will cover Chapter 1 (except for Sections 1.6 and 1.10).

Quiz 3 is a take-home quiz (download here). Please complete and turn quiz with Homework Assignment #3 in class on Monday, February 27.

Homework Assignmet #4 (Complete before Exam 1. Assignment will not be collected.)

## Week 4 (Week of February 13)

Helpful reading: Sections 1.5 and 1.7

We did not have class on Monday (CUNY closed).

Wednesday's Class: Introduced the notion of linear independence, discussed examples, and gave a condition for the columns of a matrix to be linearly independent.

Homework Assignment #3 (Due Wednesday, February 22). -Homework will be collected on Monday, February 27.

Next week's schedule: No class on Monday (CUNY closed), class on Tuesday (CUNY follows Monday schedule), class will not meet on Wednesday (I will post a video).

Quiz 3 will be a take-home quiz that I send out on Wednesday, February 22. -the quiz will be collected on Monday, February 27.

## Week 3 (Week of February 6)

Helpful reading: Sections 1.3, 1.4, and 1.5.

Monday's Class: Further discussed vectors, introduced linear combinations, vector equations, spanning sets, dot product, and the product of a matrix and a vector.

HW 1 is due on Wednesday and Quiz 1 is on Wednesday

Links to applets used in class: vector in 3-space, scalar multiplication, span of 2 vectors in 2-space, span of 3 vectors in 3-space. All applets taken from the free online textbook Interactive Linear Algebra.

Wednesday's Class: Explained how to move between linear systems, vector equations, and matrix equations. Introduced column spaces and the relation to the consistency of a matrix equation. Gave a condition on a matrix A in terms of pivot positions to determine when the Ax=b is consistent for every b. Discussed homogeneous linear systems.

Homework Assignment #2 (Due Wednesday, February 15).

No class Monday, February 13 (CUNY is closed)

Quiz 2 will be on Wednesday, February 15.

## Week 2 (Week of January 30)

Helpful reading: Section 1.2 and 1.3.

Monday's Class: Introduced, discussed, and went over the augment martrix associated to a linear system, reduced row echelon form, and the Gauss–Jordan elimination algorithm.

Homework Assignment #1 (Due Wednesday, February 8)

No Quiz this week: first quiz will be next Wednesday.

The videos linked to on the right discuss how to find rref on a standard TI calculator and an example of someone going through the Gauss–Jordan elimination algorithm.

Wednesday's Class: Discussed how to solve a linear system using the reduced row echelon form of its associated augmented matrix. This involved introducing the notion of basic variables, free variables, and parameterized solutions. Part of our discussion was summed up in a theorem stating that a linear system is consistent if and only if the rightmost colum of the augmented matrix is not a pivot column. And, moreover, a consistent linear system either has a unique solution (no free variable case) or infinitely many solutions (free variabls exist). We finished with introducing the notion of vectors, vector addition, and scalar multiplication.

## Week 1 (Week of January 23)

Helpful reading: Section 1.1 of text.

Wednesday's Class: Introduced Google's PageRank and worked through simple example. Defined linear system and worked through simple examples.

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.