Math 231(Last updated Wednesday, November 29 @ 10:15pm)
Instructor: Professor Nicholas Vlamis
Instructor Office: 507 Kiely
Instructor Email: email@example.com
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).
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
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.