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Pre-Class Assignments
Math 221 Linear Algebra, Fall 2024

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.

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

Week 1: Before first class meeting on Tuesday August 27

Welcome to Linear Algebra!
Systems of linear equations

To Read

No questions to submit for the first week.

Remember to fill out the Background Questionnaire linked from Canvas.

Week 2: Due Monday September 2 @ 11:59 pm

Equivalent vector equations
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?
Submit answers through Canvas

Week 3: Due Monday September 9 @ 11:59 pm

Matrices as functions

To Read

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

Pre-Class Questions

  1. 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.
  2. Suppose T: ℝ4 → ℝ5 is a linear transformation and T(x) = Ax for some matrix A. What are the dimensions of A? Why?
  3. Give an example of a function T: ℝ2 → ℝ2 that is not a linear transformation. Explain.
  4. 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.
Submit answers through Canvas

Week 4: Due Monday September 16 @ 11:59 pm

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.
Submit answers through Canvas

Week 5: Before class on Thursday September 26

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).
Think about these, but you do not need to submit answers with Exam 1 this week.

Week 6: Due Monday September 30 @ 11:59 pm

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 Canvas

Week 7: Due Monday October 7 @ 11:59 pm

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).
Submit answers through Canvas

Week 8: No assignment due

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

Week 9: Due Monday October 21 @ 11:59 pm

An invariant of vector spaces
Understanding longterm behavior

To Read

Pre-Class Questions

  1. 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.
  2. What is the point of studying Markov chains?
  3. What is the steady state vector for a stochastic matrix P?
Submit answers through Canvas

Week 10: Due Monday October 28 @ 11:59 pm

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. 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?
  2. Are the vectors \( u=\begin{bmatrix} 1\\-2\\3 \end{bmatrix} \) and \( v=\begin{bmatrix} 2\\4\\2 \end{bmatrix} \) orthogonal in ℝ3? Explain.
Submit answers through Canvas

Week 11: Due Monday November 4 @ 11:59 pm

Extending geometric intuition to higher dimensions

To Read

Pre-Class Questions

  1. Let H be the yz-plane in ℝ3.
    • What is the orthogonal complement of H in ℝ3?
    • Give an orthogonal basis for H.
  2. If \( \hat{y} \) is the orthogonal projection of the vector y onto the vector u, in what direction does the vector \( \hat{y} \) point?
  3. What is the purpose of the Gram-Schmidt process?
Submit answers through Canvas

Week 12: Before class on Thursday November 14

Finding the closest vector

To Read

Pre-Class Questions

  1. Does every system Ax=b have a solution x?
  2. Does every system Ax=b have a least-squares solution \( \hat{x} \)? Explain.
Think about these, but you do not need to submit answers with Exam 1 this week.

Week 13: Due Monday November 18 @ 11:59 pm

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.
  3. Give an example of a 3x3 matrix A that is symmetric and a 3x3 matrix B that is not symmetric.
Submit answers through Canvas

Week 14: Due Monday November 25 @ 11:59 pm

A factorization of m x n matrices

To Read

Pre-Class Questions

  1. 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?
  2. If A is any mxn matrix, what special property does the matrix ATA have?
Submit answers through Canvas

Week 15: Before class on Tuesday December 3

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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