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Pre-Class Assignments
Math 221 Linear Algebra, Fall 2021

This page uses MathJax to display mathematical notation, so please let me know if any part isn't clear.

All section numbers refer to the 3rd and 4th editions of the text, Linear Algebra and Its Applications by Lay.

There are other resources that you might find useful for reference, including

Be sure to check back, because this page will be updated often during the semester.

Week 1: Due Monday August 30 @ midnight

Welcome to Linear Algebra!
Systems of linear equations and equivalent vector equations

To Read

Read these sections, but no questions to submit since this is the first week of class.
Do remember to fill out the Background Questionnaire linked from onCourse.

Week 2: Due Monday September 6 @ midnight

Describing all solutions to a system

To Read

Pre-Class Questions

  1. Let \( A=\begin{bmatrix} 1& -3 \\ 2& 1\\ -1& 5 \end{bmatrix} \) and \( b = \begin{bmatrix} -3 \\ 8 \\ 7 \end{bmatrix} \)
    • Does Ax=b have a solution for this specific value of b? Explain.
    • Does Ax=b have a solution for every b ∈ ℝ3? Explain.
  2. If the system Ax=b is consistent and Ax=0 has a non-trivial solution, how many solutions does Ax=b have?
  3. Let \( u=\begin{bmatrix} 1\\3\\2 \end{bmatrix}, v=\begin{bmatrix} -1\\1\\-2 \end{bmatrix}, w=\begin{bmatrix} -1\\11\\2 \end{bmatrix} \). Is the set of vectors {u, v, w} linearly independent or linearly dependent? Explain.
  4. Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through onCourse

Week 3: Due Monday September 13 @ midnight

Matrices as functions

To Read

Pay special attention to the statements of Theorems 11 and 12 in Section 1.9.

Pre-Class Questions

  1. Suppose T: ℝ4 → ℝ5 is a linear transformation and T(x) = Ax for some matrix A. What are the dimensions of A? Why?
  2. Give an example of a function T: ℝ2 → ℝ2 that is not a linear transformation. Explain.
  3. Give the standard matrix A for the linear transformation T: ℝ2 → ℝ2 that rotates about the origin by an angle of π/3 radians counterclockwise.
  4. Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through onCourse

Week 4: Due Monday September 20 @ midnight

Multiplying and inverting matrices

To Read

Pre-Class Questions

  1. Let \( A = \begin{bmatrix} 1 & 2 & -1 \\ -3 & -5 & -3 \end{bmatrix} \) and \( B = \begin{bmatrix} -1&3 \\ 2 & -1 \\ 0 & 2 \end{bmatrix} \)
    Compute the following products or explain why it is impossible: AB, BA, BAT
  2. Let \( B = \begin{bmatrix} 1&0&1 \\ 0&1&2 \\0&2&5 \end{bmatrix} \). Find B-1
  3. If A is an invertible matrix, are the rows of A a linearly independent set or a linearly dependent set? Explain.
  4. Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through onCourse

Week 5: Due Monday September 27 @ midnight

Using matrices to shift, rotate, and skew graphics; The determinant function

To Read

Pre-Class Questions

  1. Why are homogenous coordinates used in computer graphics?
  2. Let \( A=\begin{bmatrix} 3&2\\-4&1 \end{bmatrix} \). Find det(A).
  3. Let \( B=\begin{bmatrix} 3&0&1 \\0&-1&2 \\0&0&5 \end{bmatrix} \). Find det(B).
Submit answers through onCourse

Week 6: Due Monday October 4 @ midnight

Identifying underlying structural similarities

To Read

Pre-Class Questions

  1. Give an example of a subset of ℝ2 that is a subspace of ℝ2. Explain.
  2. Give an example of a subset of ℝ2 that is not a subspace of ℝ2. Explain.
  3. If the columns of A are linearly independent, what is Nul(A)? Why?
  4. If A is 6 x 9 with 6 pivots, what is Col(A)? Why?
Submit answers through onCourse

Week 7: No assignment due

No class meetings this week due to Fall Break and MAP Day.

Week 8: Due Monday October 18 @ midnight

Minimal generating sets

To Read

Pre-Class Questions

    Let \( A = \begin{bmatrix} 1&2&5 \\ 3&5&14 \\ 1&3&6 \end{bmatrix} \)
  1. Do the columns of A form a basis for ℝ3? Explain.
  2. Give a basis for Col(A).
  3. Give a basis for Nul(A).
  4. Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through onCourse

Week 9: Due Monday October 25 @ midnight

Adjusting your reference; An invariant of vector spaces

To Read

Pre-Class Questions

  1. Let \( b_1 = \begin{bmatrix} 2\\-1 \end{bmatrix}, b_2 = \begin{bmatrix} -1\\3 \end{bmatrix}, \mathcal{B}=\{b_1,b_2\}, \) and \( x = \begin{bmatrix} 3\\1 \end{bmatrix} \)
    • Show that \( \mathcal{B} \) is a basis for ℝ2.
    • Find \( [x]_{\mathcal{B}} \), the coordinates of \( x \) relative to the basis \( \mathcal{B} \).
  2. Give an example of a 3 x 3 matrix A where the dimension of Col A is 2 and the dimension of Row A is 3, or else explain why no such matrix exists.
  3. Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through onCourse

Week 10: Due Monday November 1 @ midnight

Understanding longterm behavior; Directions fixed by matrix functions

To Read

You can skim the part of Section 5.2 labeled Determinants. One path through the text is to skip Chapter 3, so that is why this information is included here.

Pre-Class Questions

  1. What is the point of studying Markov chains?
  2. What is the steady state vector for a stochastic matrix P?
  3. Let \( A = \begin{bmatrix} 7&2 \\ -4&1 \end{bmatrix} \)
    • Verify that \( x=\begin{bmatrix} -1\\2 \end{bmatrix} \) is an eigenvector for A with corresponding eigenvalue \( \lambda=3 \).
    • What is the characteristic equation for A?
  4. Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through onCourse

Week 11: Due Monday November 8 @ midnight

Extending geometric intuition to higher dimensions

To Read

Pre-Class Questions

  1. Are the vectors \( u=\begin{bmatrix} 1\\-2\\3 \end{bmatrix} \) and \( v=\begin{bmatrix} 2\\4\\2 \end{bmatrix} \) orthongonal in ℝ3? Explain.
  2. Let H be the yz-plane in ℝ3.
    • What is the orthogonal complement of H in ℝ3?
    • Give an orthogonal basis for H.
  3. If \( \hat{y} \) is the orthogonal projection of the vector y onto the vector u, in what direction does the vector \( \hat{y} \) point?
Submit answers through onCourse

Week 12: Due Monday November 15 @ midnight

Finding the closest vector

To Read

Pre-Class Questions

  1. Let \( y=\begin{bmatrix} 1\\3\\5 \end{bmatrix} \) in ℝ3 and let W be the xy-plane in ℝ3. Find the orthogonal projection of y onto W.
  2. What is the purpose of the Gram-Schmidt process?
  3. Does every system Ax=b have a solution x? Does every system Ax=b have a least-squares solution \( \hat{x} \)? Explain.
Submit answers through onCourse

Week 13: Due Monday November 22 @ midnight

A factorization of certain square matrices

To Read

Pre-Class Questions

  1. What is the point of finding a diagonalization of a matrix?
  2. If A is 4 x 4 with eigenvalues 1, 2, 0, and 3, is A diagonalizable? Explain.
Submit answers through onCourse

Week 14: Due Monday November 29 @ midnight

A factorization of m x n matrices

To Read

Pre-Class Questions

  1. Give an example of a 3x3 matrix A that is symmetric and a 3x3 matrix B that is not symmetric.
  2. If A is a symmetric matrix and \( \vec{v_1} \) and \( \vec{v_2} \) are eigenvectors of A coming from different eigenvalues, how are \( \vec{v_1} \) and \( \vec{v_2} \) related geometrically?
  3. If A is any mxn matrix, what special property does the matrix ATA have?
Submit answers through onCourse

Week 15: Due Monday December 6 @ midnight

The power of the singular value decomposition

To Read

There are lots of nice applications in this book, and it's available through the library as an ebook. It's definitely worth checking out.

No questions to submit this week.


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