Pre-Class AssignmentsMath 221 Linear Algebra, Fall 2023

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: Due Monday August 28 @ 11:59 pm

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

• 1.1 Systems of Linear Equations
• 1.2 Row Reduction and Echelon Forms
• 1.3 Vector Equations
Read these sections, but no questions to submit since this is the first week of class.
Remember to fill out the Background Questionnaire linked from Canvas!

Week 2: Due Monday September 4 @ 11:59 pm

Describing all solutions to a system

• 1.4 The Matrix Equation Ax=b
• 1.5 Solution Sets of Linear Systems
• 1.7 Linear Independence

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?

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

Matrices as functions

• 1.8 Introduction to Linear Transformations
• 1.9 The Matrix of a Linear Transformation

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?

Week 4: Due Monday September 18 @ 11:59 pm

Multiplying and inverting matrices

• 2.1 Matrix Operations
• 2.2 Inverse of a Matrix
• 2.3 Characterizations of Invertible Matrices

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?

Week 5: Due Monday September 25 @ 11:59 pm

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

• 2.7 Applications to Computer Graphics
• 3.1 Introduction to Determinants
• 3.2 Properties of Determinants

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 October 2 @ 11:59 pm

Identifying underlying structural similarities

• 4.1 Vector Spaces and Subspaces
• 4.2 Null Spaces, Column Spaces, and Linear Transformations

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?

Week 7: No assignment due

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

Week 8: Due Monday October 16 @ 11:59 pm

Minimal generating sets

• 4.3 Linearly Independent Sets; Bases

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?

Week 9: Due Monday October 23 @ 11:59 pm

• 4.4 Coordinate Systems
• 4.5 The Dimension of a Vector Space
• 4.6 Rank

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?

Week 10: Due Monday October 30 @ 11:59 pm

Understanding longterm behavior; Directions fixed by matrix functions

• 4.9 Applications to Markov Chains
• 5.1 Eigenvectors and Eigenvalues
• 5.2 The Characteristic Equation

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?

Week 11: Due Monday November 6 @ 11:59 pm

Extending geometric intuition to higher dimensions

• 6.1 Inner Product, Length, and Orthogonality
• 6.2 Orthogonal Sets

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?

Week 12: Due Monday November 13 @ 11:59 pm

Finding the closest vector

• 6.3 Orthogonal Projections
• 6.4 The Gram-Schmidt Process
• 6.5 Least-Squares Problems

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.

Week 13: Due Monday November 20 @ 11:59 pm

A factorization of certain square matrices

• 5.3 Diagonalization

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.

Week 14: Due Monday November 27 @ 11:59 pm

A factorization of m x n matrices

• 7.1 Diagonalization of Symmetric Matrices
• 7.4 The Singular Value Decomposition

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?
4. Have you and your partner met to discuss the Problem Set? How much progress have you made?