Math 22: Linear Algebra with Applications

Spring 2026

Homework Assignments


  • The latest homework will always appear on top.

  • Late Homework will not be accepted. If you must miss a class, it is your responsibility to submit all homework on time.
  • Make sure that in your homework you indicate everyone that helped you in the homework or anyone with whom you collaborated. For example, write "Tutorial" or "Office Hours", if you received help from these sources.

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

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

    Week 10
    Saturday, June 6:
    • Topic: Final Exam at 11:30 AM in Wilder 104.
    Wednesday, June 3:
    • Topic: Review Problems.
    • All webwork will close tonight by 11PM.
    Monday, June 1:
    • Topic: The Principal Component Analysis (PCA)
      Introduction to Principal Component Analysis. The applications of PCA will differ per section, sample applications include data analysis and face recognition algorithms.
    • No new Webwork: finish Webwork #26 and #27 if you have not done them.
      • Reading Homework: Review relevant sections in your book to prepare for the exam.
    Week 9
    Friday, May 29:
    • Topic: Applications to image processing, JPEG
      we discuss a change of basis algorithm used to store images and how it applies to image compression.
    Wednesday, May 27:
    • Topic: Applications of Singular value decomposition (SVD)
      We continue discussion on the SVD and applications such as image compression, noise reduction, and data analysis.
    • WebWork #26 Sec. 7.4 (due Friday at 11PM)
      • Reading Homework: Read Section 7.5 (due Friday at the beginning of class)
    • Suggested Written Homework: 7.4 # 7, 12, 15, 18 (from your book, go to files in Canvas for the problems) (This will not be collected)
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    X-hour, May 26:
    • Topic: The Singular Value Decomposition (SVD)
      Introduction to Singular Value Decomposition.
    • WebWork #25 Sec. 7.4 (due Friday at 11PM)
      • Reading Homework: Read Section 7.5 (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 22:
    • 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 Tues. May 26 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Wednesday, May 20:
    • 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 #23 Sec. 4.9 and 5.8 (due Friday before class)
      • Reading Homework: Read Section 7.4 (due Monday at the beginning of class)
    • Written Homework: Homework Set #8 (due Tues. May 26 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, May 18
    • Topic: Orthogonally diagonalizable and symmetric matrices
      Today we discuss symmetric matrices and orthogonal diagonalization. Including the spectral decomposition theorem.
    • WebWork #22 Sec. 7.1 (due Friday May 22 at 11PM)
    • Reading Homework: Read Section 4.9 and 5.8 (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 15:
    • 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 #21 6.4 (due Wed. at 11PM)
    • Reading Homework: Read Sect. 7.1 (due Monday at the beginning of class)
    • Written Homework: Homework Set #7 (due Tues. May 19 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Thursday, May 14:
    • Midterm #2 in 200 Life Sciences (Last Name starts A-K) and 201 Life Sciences (Last Name starts L-Z) at 7:00 - 9:00 PM.
    Wednesday, May 13:
    • Topic: Orthogonal Projections
      Today we will be studying orthogonal projections, the orthogonal decomposition theorem and the best approximation theorem.
    • WebWork #20 Sec. 6.2 and 6.3 (due Friday at 11PM)
      • Reading Homework: Read Section 6.4 and watch the videos (due Friday at the beginning of class)
    • Written Homework: Homework Set #7 (due Tues. May 19 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, May 11:
    • 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 #19 Sec. 6.1 (due Friday at 11PM)
    • Reading Homework: Read Section 6.2 and 6.3 (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 8:
    • 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 at 11pm)
    • Reading Homework: Read Sect. 6.1 and 6.2 (due Monday at the beginning of class)
    • Written Homework: Homework Set #6 (due Tues. May 12 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Wednesday, May 6:
    • Topic: The characteristic equation and the characteristic polynomial of a matrix.
      We will learn how to compute the eigenvalues and the definition of similar matrices. And how similar matrices connect 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 at 11pm)
      • Reading Homework: Read Section 5.3 and 5.4 (due Friday at the beginning of class)
    • Written Homework: Homework Set #6 (due Tues. May 12 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, May 4:
    • Topic: Eigenvalues, Eigenvectors and Eigenspaces
      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 #16 Sec. 5.1 (due Wednesday at 11pm)
    • 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, May 1:
    • 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 #14 Sec. 3.1 and 3.2 (due Monday 11PM)
    • Reading Homework: Read Sect. 5.1 and 5.2 (due Monday at the beginning of class)
    • Written Homework: Homework Set #5 (due Tues. May 5 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Wednesday, April 29:
    • 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 #14 Sec. 4.7 and 5.4 (due Friday 11PM)
      • Reading Homework: Read Section 3.1 and 3.2 (due Friday at the beginning of class)
    • Written Homework: Homework Set #5 (due Tues. May 5 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, April 27:
    • 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 #13 Sec.4.4 and 5.4 (due Wednesday 11PM)
    • Reading Homework: Read Section 4.4, 4.7 and 5.4 (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 24:
    • Topic: Coordinates, Dimension, and Rank
      Sec. 2.9 Coordinates, dimension and rank Today we define the coordinate map, define dimension and rank.
    • WebWork #12 Sec. 2.9 (due Monday at 11PM)
    • Reading Homework: Read Sect. 4.4, 4.7 and 5.4 (due Monday at the beginning of class)
    • Written Homework: Homework Set #4 (due Tues. April 28 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Thursday, April 23:
    • Midterm #1 in Life Science 100 Oopik Auditorium at 7:00 - 9:00 PM.
    Wednesday, April 22
    • 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 #11 Sec. and 4.3 (due Friday before class)
      • Reading Homework: Read Section 2.9 (due Friday at the beginning of class)
    • Written Homework: Homework Set #4 (due Tues. April 28 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, April 20
    • 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 #10 Sec. 2.3 (due Wednesday 11PM)
    • Reading Homework: Read Section 4.3 (due Wednesday at the beginning of class)
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Week 3
    Friday, April 17:
    • 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 #9 Sec. 2.1 (due Monday at 11PM)
    • Reading Homework: Read Sect. 2.3 (due Monday at the beginning of class)
    • Written Homework: Homework Set #3 (due Tues. April 21 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Wednesday, April 15:
    • 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 #8 Sec. and 2.1 (due Friday at 11PM)
      • Reading Homework: Read Section 2.2 (due Friday at the beginning of class)
    • Written Homework: Homework Set #3 (due Tues. April 21 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, April 13:
    • 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 #7 Sec. 4.2 and 1.9 (due Wednesday at 11PM)
    • Reading Homework: Read Section 1.9 and 2.1 (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 10:
      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 #6 Sec. 4.1 and 4.2 (due Monday at 10PM)
    • Reading Homework: Read Sect. 4.2 and 1.9 (due Monday at the beginning of class)
    • Written Homework: Homework Set #2 (due Tues. April 14 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Wednesday, April 8:
    • Topic: Linear Independence
      Today we will learn about linear independent sets and connections to linear systems.
    • WebWork #5 Sec. 1.7 (due Friday by 11 PM)
      • Reading Homework: Read Section 4.1-4.2 (due Friday at the beginning of class)
    • Written Homework: Homework Set #2 (due Tues. April 14 at 11 PM EST or before)
      Submit the homework using Gradescope.
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, April 6:
    • 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 #4 Sec. 1.4 and 1.5 (due Wednesday by 11 PM)
    • Reading Homework: Read Section 1.7 (due Wednesday at the beginning of class)
    Week 1
    Friday, April 3:
    • 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 3: Sec. 1.3 and 1.4 (due Monday by 11PM)
    • Reading Homework: Read Sect. 1.5 (due Monday at the beginning of class)
    • Written Homework: Homework Set #1 (due Tues. April 7 at 11PM EST or before)
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Wednesday, April 1:
    • 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 2: Sec. 1.2 (due Friday by 11PM)
      • Reading Homework: Read Section 1.3-1.4 (due Friday at the beginning of class)
    • Written Homework: Homework Set #1 (due Tues. April 7 at 11PM EST or before)
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas.
    Monday, March 30:
    • 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 1: Sec. 1.1 (due Wednesday by 11PM)
    • Reading Homework: Read Section 1.1 and 1.2 (due Wednesday at the beginning of class)
    • Discussion Homework: Make sure to visit and post in the Discussion page for Canvas (due Sunday by 10 PM).


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