# Math 22: Linear Algebra with Applications

Spring 2021

## Homework Assignments

• The latest homework will always appear on top.

• If you are interested in learning to typeset mathematics here is a list of resources compiled by Rebecca Weber on LaTex

• Late Homework will not be accepted.
• Make sure that in your homework you indicate everyone that helped you in the homework or anyone with whom you collaborated.

WebWork problems: These problems are very important. We will discuss most of them in class. Please be sure that you understand these problems thoroughly as you might be asked to present some of these problems in class.

Written assignments: These problems will graded in detail and are due once a week on Wednesdays. You should make sure that they are well-written.

Participation assignments: Every week a student will receive a participation grade, each instructor will inform students how students will be evaluated.

Week 10
 Sunday, June 6: Final Exam
 Wednesday, June 2: Topic: Review We will review the main ideas in Linear Algebra!
 Monday, May 31: Memorial Day! Enjoy the day off
Week 9
 Friday, May 28: Topic: The Principle Component Analysis (PCA ) and applications Applications will differ per section, sample applications include data analysis and face recognition algorithms. WebWork Review (No due date) Suggested Homework: Suggested Written Homework: Sect. 7.4 # 7, 12, 15, 18 (This will not be collected) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, May 26: Topic: Image compression and the JPEG algorithm We continue discussion on image compression, we discuss a change of basis algorithm used to store images and how it applies to image compression. WebWork Sec. 7.4 (due Friday before class) Reading Homework: Read Section 7.4 and watch the videos (due today at the beginning of class) Suggested Written Homework:7.4 # 7, 12, 15, 18 (This will not be collected) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, May 24: Topic: The Singular Value Decomposition and applications Introduction to Singular Value Decomposition and application to data compression. WebWork Sec. 7.4 (due Wednesday before class) Reading Homework: Read Section 7.4 and watch the videos (due today at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 8
 Friday, May 21: Topic: The Page Rank Algorithm Today we give an application of Markov chains to search engines. WebWork Review (due Wednesday before class) Reading Homework: Read Section 7.4 (due Wednesday at the beginning of class) Written Homework: Homework Set #8 (due Wed. May 12 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, May 19: Topic: Markov chains and the power method Today we will discuss the basics of Markov chains and the power method for approximating eigenvalues and eigenvectors. WebWork Sec. 4.9 and 5.8 (due Friday before class) Reading Homework: Read Section 7.4 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #8 (due Wed. May 26 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, May 17: Topic: Orthogonally diagonalizable and symmetric matrices Today we discuss symmetric matrices and orthogonal diagonalization. Including the spectral decomposition theorem. WebWork Sec. 7.1 (due Wednesday before class) Reading Homework: Read Section 4.9 and 5.8 and watch the videos (due Monday at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 7
 Friday, May 14: Topic: Orthogonal Projections and the Gram-Schmidt Process Today we will learn how to compute orthogonal bases using the orthogonal decomposition theorem and we will talk about an application of the best approximation theorem to finding least square solutions. WebWork 6.4 (due Monday before class) Reading Homework: Read Sect. 7.1 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #7 (due Wed. May 19 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, May 12: Topic: Orthogonal Projections Today we will be studying orthogonal projections, the orthogonal decomposition theorem and the best approximation theorem. WebWork Sec. 6.2 and 6.3 (due Friday before class) Reading Homework: Read Section 6.4 and watch the videos (due Friday at the beginning of class) Written Homework: Homework Set #7 (due Wed. May 19 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, May 10: Topic: The Inner Product, length, orthogonal complements and orthogonality Today we will be learning about the inner product and how this leads to definitions of length, distance, angle and orthogonality. We define orthogonal complement and define Row A which is the orthogonal complement of Nul A. We will define what it means for a basis to be orthogonal. WebWork Sec. 6.1 (due Wednesday before class) Reading Homework: Read Section 6.2 and 6.3 and watch the videos (due Monday at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 6
 Friday, May 7: Topic: Diagonalization and connections to linear transformations. We will learn how to diagonalize a matrix and understand how this is a factorization problem. We will also make connections to linear transformations. WebWork Sec. 5.3 and 5.4 (due Monday before class) Reading Homework: Read Sect. 6.1 and 6.2 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #6 (due Wed. May 12 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, May 5: Topic: The characteristic equation and the characteristic polynomial of a matrix. We will learn about similar matrices and how it connects to the problem of finding eigenvalues and eigenvectors. We will learn how to compute the eigenvalues and about multiplicities of eigenvalues. WebWork Sec. 5.2 (due Friday before class) Reading Homework: Read Section 5.3 and 5.4 and watch the videos (due Friday at the beginning of class) Written Homework: Homework Set #6 (due Wed. May 12 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, May 3: Topic: Computing the Eigenvectors of a matrix Today we will be learning how to compute the eigenvectors when we are given the eigenvalues. We will also show that eigenvectors corresponding to distinct eigenvalues are linearly independent. WebWork Sec. 5.1 (due Wednesday before class) Reading Homework: Read Section 5.2 and watch the videos (due Wednesday at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 5
 Friday, April 30: Topic: The determinant of a matrix and its properties We will describe two ways to compute the determinant of a matrix and study its properties. WebWork Sec. 3.1 and 3.2 (due Monday before class) Reading Homework: Read Sect. 5.1 and 5.2 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #5 (due Wed. May 5 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, April 28: Topic: Matrix of a linear transformation, change of coordinates matrix and the matrix of the composition of linear tranformations. We will continue the discussion from Monday, we will learn how to compute the change of coordinates matrix and the matrix of the composition. WebWork Sec. 4.7 and 5.4 (due Friday before class) Reading Homework: Read Section 3.1 and 3.2 and watch the videos (due Friday at the beginning of class) Written Homework: Homework Set #5 (due Wed. May 5 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, April 26: Topic: The matrix of a linear transformation T : V --> W and the change of coordinates Matrix. We will define an m x n matrix for any linear tranformation from an n-dimensional vector space V to an m-dimensional vector space W. We will obtain the change of coordinates matrix as a special case when T is the identity transformation. WebWork Sec.4.4 and 5.4 (due Wednesday before class) Reading Homework: Read Section 4.4, 4.7 and 5.4 and watch the videos (due Wednesday at the beginning of class ) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 4
 Friday, April 23: Topic: Inverse of a Matrix Sec. 2.9 Coordinates, dimension and rank Today we define the coordinate map, define dimension and rank. WebWork Sec. 2.9 (due Monday before class) Reading Homework: Read Sect. 4.4, 4.7 and 5.4 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #4 (due Wed. April 28 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, April 21: Topic: Linear independent sets and bases Sec. 4.3 Today we discuss linear independent sets for any vector space and define bases for any vector space. We discuss how to get bases from spanning sets and how to compute bases for Nul A and Col A. WebWork Sec. and 4.3 (due Friday before class) Reading Homework: Read Section 2.9 and watch the videos (due Friday at the beginning of class) Written Homework: Homework Set #4 (due Wed. April 28 at 10PM EST or before) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, April 19: Topic: Inverse Matrix Theorem and connections to linear transformations Sec. 2.3: Today we learn the Inverse Matrix Theorem (IMT) we discuss the composition of linear transformations, define the inverse of an invertible linear transformation. WebWork Sec. 2.3 (due Wednesday before class) Reading Homework: Read Section 4.3 and watch the videos (due Monday at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 3
 Friday, April 16: Topic: Inverse of a Matrix Sec. 2.2 Today we will learn an algorithm for finding the inverse of a matrix based on the elementary row operations. We will also see that elementary row operations have matrix representations. We will also discuss the linear system Ax=b when A is invertible. WebWork Sec. 2.1 (due Monday before class) Reading Homework: Read Sect. 2.3 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #3 (due Wed. April 21 at 10PM EST or before ) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, April 14: Topic: One-to-one and onto linear transformations and Matrix operations Sec. 1.9 and 2.1 Today we will study one-to-one and onto linear trasformations and see how this connects to the fundamental questions that we have been discussing in class. WebWork Sec. and 2.1 (due Friday before class) Reading Homework: Read Section 2.2 and watch the videos (due Friday at the beginning of class) Written Homework: Homework Set #3 (due Wed. April 21 at 10PM EST or before) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, April 12: Topic: Linear Transformations Sec. 4.2 and 1.9: Today we will learn about linear transformations, the kernel and the range of a linear transformation and how these relate to Nul A and Col A for a matrix A. WebWork Sec. 4.2 and 1.9 (due Wednesday before class) Reading Homework: Read Section 1.9 and 2.1 and watch the videos (due Monday at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 2
 Friday, April 9: Topic: Introduction to Vector Spaces, null space, column space. Today we will learn the general framework for what we have been doing. This will help us apply more general theorems and get a better understanding of what we have been doing. WebWork Sec. 4.1 and 4.2 (due Monday before class) Reading Homework: Read Sect. 4.2 and 1.9 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #2 (due Wed. April 14 at 10PM EST or before) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, April 7: Topic: Linear Independence Today we will learn about linear independent sets and connections to linear systems. WebWork Sec. 1.7 (due Friday before class) Reading Homework: Read Section 4.1-4.2 and watch the videos (due Friday at the beginning of class) Written Homework: Homework Set #2 (due Wed. April 14 at 10PM EST or before) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, April 5: Topic: The Matrix Equation Ax=b and its solutions Today we will continue discussing the matrix equation Ax=b and introduce homogeneous systems and relate solutions to homogeneous systems to solutions of Ax=b in general. WebWork Sec. 1.5 (due Wednesday before class) Reading Homework: Read Section 1.7 and watch the videos (due Monday at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
Week 1
 Friday, April 2: Topic: Vector Equations; Matrix Equations Today we will learn how to interpret a system of linear equations both as a vector equation or as a matrix equation. WebWork Sec. 1.3 and 1.4 (due Monday before class) Reading Homework: Read Sect. 1.5 and watch the videos (due Monday at the beginning of class) Written Homework: Homework Set #1 (due Wed. April 7 at 10PM EST or before) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Wednesday, March 31: Topic: Row Reduction and Echelon Forms Today we will learn about solving systems of linear equations and converting one system to another system with the same solution set. WebWork Sec. 1.2 (due Friday before class) Reading Homework: Read Section 1.3-1.4 and watch the videos (due Friday at the beginning of class) Written Homework: Homework Set #1 (due Wed. April 7 at 10PM EST or before) Submit the homework using Gradescope. Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
 Monday, March 29: Topic: Linear Systems of Equations Today we will have an introduction to the course as well as introduce systems of linear equations and solutions to these systems. WebWork Sec. 1.1 (due Wednesday before class) Reading Homework: Read Section 1.2 and watch the videos (due Wednesday at the beginning of class) Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.

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