# Math 231

(Last updated Wednesday, November 29 @ 10:15pm)## 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 14 (Week of November 27)
Suggested Reading: Sections 6.4 and 6.5.

Homework Assignment #12. Due Wednesday, December 6.

Monday's class: Established the Gram–Schmidt process for finding an orthonormal basis for a subspace of R^n. Proved the QR-Decomposition theorem, and as an application, explained how to use QR-decomposition to approximate eigenvalues.

Wednesday's class: Defined least-squares solutions to linear systems. Proved least-squares solutions always exist and characterized when they are unique. Gave a formulat for the least-squares solution using the QR-decomposition theorem. Explained how linear regression is a least-squares problem.

- Week 13 (Week of November 20)
Suggested Reading: Sections 6.1, 6.2, and 6.3.

Homework Assignment #11. Due Wednesday, November 30.

Monday's class: Introduced the notion of orthogonal/orthonormal bases. Gave a formula for computing coordinates with respect to an orthogonal basis. Went over the orthogonal projection theorem.

- Week 12 (Week of November 13)
Suggested Reading: Sections 5.9, 10.1, and 10.2 (Chapter 10 can be found at bit.ly/2nj1HhO).

Relevant Links to Monday's discussion: OpenAI's embeddings, Cosine similarity (Wikipedia), Markov Chain (Wikipedia)

Homework Assignment #11. Due Wednesday, November 30.

Wednesday Office Hours are cancelled.

Monday's class: Introduced Markov chains and stochastic matrices. Went over some examples and proved some essential facts about stochastic matrices.

Wednesday's class: Finished discussion of Markov chains and defined Google PageRank as the stable the vector of a Markov chain. Introduced the notion of orthogonality and the orthogonal complement of a subspace.

- Week 11 (Week of November 6)
Suggested Reading: Sections 6.1

Exam week

No homework. No quiz on Wednesday

Monday's class: Recalled the definition of the dot product and introduced the notions of norm, distance, and angle in R^n. Gave an application to search using the idea of word space. Proved the Cauchy-Schwarz inequality and the triangle inequality.

- Week 10 (Week of October 30)
Suggested Reading: Sections 5.1, 5.2, and 5.3

Exam 2 will be held in class on Wendesday, November 8. The exam will cover the matrial from Week 6 through Week 10. This cover the content in Homework Assignments #6 through #10. In the text, this correpsonds to Sections 2.1–2.3, 2.8, 2.9, 3.1, 3.2, 5.1–5.3. You make bring one sheet of notes containing statements of theorems and definitions.

Homework Assignment #10. This assignment will not be collect, and there will be no quiz. But it will show up on Exam 2. So you should complete it by November 8.

Monday's class: Defined eigenvalues, eigenvectors, and eigenspaces. Went over procedures to find these values/objects for a given matrix.

Wednesday's class: Introduced the notion of similar matrices. Proved that similar matrices have the same eigenvalues. Proved that a set of eigenvectors with distinct eigenvalues is linearly independent. Proved that an n x n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.

Examples of eigenfaces.

- Week 9 (Week of October 23)
Change in office hours: Monday office hours will be held 11am–12pm.

Suggested Reading: Sections 3.1 and 3.2.

Homework Assignment #9. Due Wednesday, November 1.

Monday's class: Defined the determinant. Computed determinant of triangular matrix and showed that a triangular matrix is invertible if and only its determinant is not 0. Stated a theorem showing how determinant changes under elementary row operations.

Wednesday's class: Proved that a matrix is invertible if and only if its determinant is nonzero. Discussed various properties of the determinant; in particular, we say that the determinant of a product of matrices is equal to the product of their determinants. Discussed a consequence of Cramer's Rule.

## Week 8 (Week of October 16)

There will be no class on Wednesday. However, you are responsible for all the content in Section 2.9 of the textbook. I will talk about much of it in Monday's class, but not all of it.

Reading: Section 2.9

Homework Assignment #8. Due Wednesday, October 25.

There will be a take-home quiz handed out on Wednesday. You will turn it in the following Monday.

Take-home quiz. Due Monday, October 23

Monday's Class: Proved a theorem yielding a basis for the column space of a matrix. Defined dimension and proved a theorem showing the definition makes sense. Defined the rank and nullity of a matrix, and proved the Rank-Nullity theorem.

## Week 7 (Week of October 9)

No class on Monday (CUNY closed). But we will have class on Tuesday at the regular time (CUNY follows a Monay schedule).

Office hours on Tuesday cancelled. (You can find me at the major/minor fair if you want to say hello.)

Homework Assignment #6 (Due Wednesday, October 11)

Suggested reading: Sections 2.2, 2.3, and 2.8

Homework Assignment #7. Due Wednesday, October 18. (This assignment will not be collected. There will be a take-home quiz on this assignment, as we will not meet in person next Wednesday.)

Tuesday's class: Introduced the notion of an invertible matrix. Introduced the notion of determinant for 2x2 matrices and a formula for the inverse of a 2x2 matrix. Established various properties of invertible matrices. Introduced the notion of an elementary matrix, and used elementary matrices to show that the Gauss-Jordan elimination algorithm can be used to detect if a matrix is invertible and to find the inverse of an invertible matrix.

Wednesday's class: Finished our initial discussion of invertible matrices. Introduced the notion of a subspace, discussed examples, and basic properties. Defined a basis of a subspace and computed bases of null spaces and column spaces.

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 6 (Week of October 2)

Suggested reading: Section 2.1

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 sheet of notes containing statements of theorems and definitions.

Monday's class: Defined matrix multiplication and the transpose.

## 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.