# Math 231

(Last updated Wednesday, September 27 @ 10pm)## Course Information

Instructor: Professor Nicholas Vlamis

Instructor Office: 507 Kiely

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

Class Meeting: Monday/Wednesday 3:45–5:35pm in Kiely 320

Office Hour: Monday 12–1pm and Wednesday 11am–12pm (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 5 (Week of September 25)

Suggested reading: Sections 1.9, 2.1

CUNY was closed on Monday, Septembe 25.

Homework Assignment #5. This assignment will not be collected; however, the content will appear on Exam 1.

Exam 1 will be held in class on Wednesday, October 4. The exam will cover the material from Week1 through Week 5. This corresponds to the material represented in Homework assignment #1 through Homework assignment #5. In the text, this is Sections 1.1–1.5, 1.7–1.9. You make bring one shee of notes containing statements of theorems and definitions.

Wednesday's class: Proved theorems discussing when a linear transformation is one-to-one or onto. Proved that there is no one-to-one linear transformation from R^n to R^m whenever n > m.

## Week 4 (Week of September 18)

Suggested reading: Sections 1.7, 1.8, 1.9, 2.1

Homework Assignment #4. Due Wednesday, September 20.

Monday's class: Further discussed linear independence, and established several theorems; in particular, we showed that any collection of more than n vectors in R^n is linearly dependent. We defined matrix transformations and saw several examples.

Wednesday's class: Introduced the notion of a linear transformation. Proved that every linear transformation is a matrix transformation. Discussed how to find the standard matrix of a linear transformation. Proved the angle sum identities for cosine and sine.

## Week 3 (Week of September 11)

Suggested reading: Sections 1.3, 1.4, 1.5, and 1.7.

Homework Assignment #3. Due Wednesday, September 20.

The solutions to Quiz 1 are posted above. The quiz itself is graded out of 10 points, but the total score is out of 12 points: if you turned in your homework you receive an additional two points.

Monday's class: Introduced the span of a collection of vectors. Defined the dot product and went over basic properties. Defined the product of an (m x n )-matrix and a vector in R^n in terms of linear combinations. Explained how to compute this product using dot products. Introduced the coefficient matrix and stated a theorem establishing the equivalence of solution sets between the representation of a linear system as an augmented matrix, a vector equation, and a matrix equation. Defined the column space of a matrix and went over a theorem establishing the equivalence of several conditions, namely the consistency of a matrix equation, the column space of a matrix being as large as possible, and the existence of pivot positions in every row of a matrix.

Wednesday's class: Introduced homogeneous linear systems. Showed, via a worked example, how to express the general solution of a homogeneous linear system as a span of vectors. Proved a theorem (Theorem 6) relating the general solution of Ax=b to the general solution of Ax=0. Defined linear independence for a set of vectors. Made basic observations of when a set with one or two vectors is linearly independent.

## Week 2 (Week of September 4)

CUNY was closed on Monday.

Suggested reading: Section 1.3.

Homework Assignment #2. Due Wednesday, September 13.

Quiz 2 will be given during the beginning of class on Wednesday, September 13. It will be based on the exercises in Homework Assignment #1.

Wednesday's class: Went over another example of finding the general solution to a linear system. Discussed a theorem detecting when a linear system is consistent based on the augmented matrix; the same theorem included a statement saying that a consistent linear system either has a unique solution (no free variables) or infinitely many solutions (there exists a free variable). Defined the size of a matrix and defined a (column) vector. Introduced notation for R^m, the set of (m x 1)-matrices. Discussed basic properties of vector addition and scalar multiplication. Defined linear combination and discussed how to decide if a vector is a linear combination of a set of vectors. Discussed correspondence between vector equations and linear systems.

See videos to the right for how to use TI calculator to find the reduced row echelon form of a matrix and for an additional example of working through the Gauss-Jordan elimination algorithm.

## Week 1 (Week of August 28)

Welcome to Math 231! Information for the week and a summary of classes will be added here.

Suggested reading: Section 1.1 and 1.2.

Homework Assignment #1. Due Wednesday, September 6.

Quiz 1 will be given during the beginning of class on Wednesday, September 6. It will be based on the exercises in Homework Assignment #1.

Monday's class: Discussed Google PageRank as a motivating example. Introduced linear equations, linear systems, solutions of linear systems, and augmented matrices.

Wednesday's class: Defined elementary row operations, reduced row echelon form, pivot positions, and pivot columns. Established the Gauss–Jordan elimination algorithm for finding the reduced row echelon form of a matrix. Discussed how to use this algorithm to find the general solution to a linear system. Defined basic variables, free variables, and parameterized solutions.