Weekly homework will be assigned, but it will not be collected. Instead, there will be a weekly quiz based on the prior week's assignment and lectures.
We have class on Friday, Oct. 24, as CUNY is on a Monday schedule.
Wednesday, Oct. 22: Produced an algorithm for determining whether a square matrix is invertible, and in the case it is invertible, finding the inverse. The algorithm relies on Gauss–Jordan elimination. Discussed the Inverse Matrix Theorem, going over several of its components. Disussed invertibility of linear transformations in terms of their standard matrices.
Friday, Oct. 24: Introduced the notion of a subspace and discussed several examples. Gave two subspaces associated to a matrix, namely the column space (somethign we've seen before) and the null space. Introduced the notion of a basis and showed, via an example, that the null space of a matrix A always has a basis with the same number of vectors as there are free variables in the matrix equation Ax=0.
Week 7
Reading: Sections 2.2 and 2.3
Exam 1 is on Wednesday, October 15. See Week 6 notes for details.
No class Monday, October 20, but we have a make-up class on Friday, October 24.
HW5 has been posted. Quiz 5 will be next Wednesday, Oct. 22.
Tuesday, Oct. 14 (Day 10): Defined elementary matrix. Explained how elementary row operations can be accomplished via left multiplication of an elementary matrix. Proved that a square matrix is invertible if and only if it is row equivalent to an identity matrix. The proof gives an algorithm for verifying invertibility and computing the inverse, which we will discuss in detail next time.
Wednesday, Oct. 15 (Day 11): Exam 1
Week 6
Reading: Sections 2.1 and 2.2
Exam 1 is on Wednesday, October 15. You may bring one sheet of notes—containing only statements of theorems and definitions—to the exam.
Exam 1 will cover the following sections of the book. From Chapter 1, the exam will cover Sections 1, 2, 3, 4, 5, 7, 8, 9, and from Chapter 2 only section 1. This covers the first six weeks of class and the first four homework assignments.
There is no class on Monday, Oct. 13, but we do have class on Tuesday, Oct. 14 at the usual time and place.
Monday, Oct. 6 (Day 8): Began with expressing two-dimensional rotations as matrices. Proved the angle-sum formula from trigonometry. Began discussing matrix operations, including addition, scalar multiplication, and the definition of matrix multiplication.
Wednesday, Oct. 8 (Day 9): Derived formula for matrix multiplication. Discussed properties of matrix multiplication. Defined invertible matrix. Defined the determinant for a 2x2 matrix and gave a condition for the invertibility of a 2x2 matrix in terms of the determinant, as well as a formula for the inverse when in it exists.
Week 5
Reading: Sections 1.8 and 1.9
Monday, Sept. 29 (Day 7): Introduced linear and matrix transformations and proved they are one and the same! Gave some standard examples from geometry of linear transformations.
Week 4
Reading: Sections 1.5, 1.7
No office hours on Wednesday.
To simplify some of HW3, you might find it helpful to use a calculator (or another source) to automate Gauss–Jordan elemination. Here is a link to a video of someone showing how to use a TI-84 to row reduce a matrix: https://youtu.be/zhypLK9nK80
There are no classes Monday through Wednesday next week at CUNY, so we will not meet again until Monday, Sept. 29.
Monday, Sept. 15 (Day 5): Established several equivalent conditions for the matrix equation Ax=b to be consistent for all vectors b. Introduced homogeneous linear systems and showed how to write their solutions as a span of vectors. Proved that the solution set of Ax=b, when consistent, is a translation of the solution set of Ax=0.
Wednesday, Sept. 17 (Day 6): Showed how to write the solution of a linear system of the form Ax=b as a translation of a span of vectors. Introduced the notion of linear independence and established several basic facts.
Week 3
Reading: Section 1.2–5
Our first quiz will be on Wednesday. The quiz will be given during the final 15 minutes of class and will be based on HW1.
Monday, Sept. 8 (Day 3): Discussed how to write the general solution of a linear equation using the reduced row echelon form of the associated augmented matrix, leading and free variables, and parameters. Introduced vectors, linear combinations, and vector equations. Observed that vector equations encode a linear system.
Wednesday, Sept. 10 (Day 4): Introduced the notion of the span of a set of vectors, introduced the dot product, and defined the notion of multiplication of an m-by-n matrix and an n-dimensional vectors. We then discussed the equivalence of linear systems, vector equations, and matrix equations. Introduced the column space of a matrix and showed that the matrix equation Ax=b has a solution if and only if b is in the column space of A.
Week 2
Reading: Section 1.1 and 1.2
Office hours on Wednesday, September 3 will be moved to 3:30–4:30pm (due to a department meeting).
Wednesday, Sept. 3 (Day 2): Introduced augmented matrices and the Gauss—Jordan elimination algorithm.
Week 1
Reading: Section 1.1
The first homework assignment will be assigned next week.
Wednesday, Aug. 27 (Day 1): Went over two motivating examples: imgae compression and Google PageRank. Defined a linear system, did an example, and visualized solution.