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
To Read
 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
To Read
 1.4 The Matrix Equation Ax=b
 1.5 Solution Sets of Linear Systems
 1.7 Linear Independence
PreClass Questions

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 ∈ ℝ^{3}? Explain.

If the system Ax=b is consistent and Ax=0 has a nontrivial solution, how many solutions does Ax=b have?

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.
 Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through Canvas
Week 3: Due Monday September 11 @ 11:59 pm
Matrices as functions
To Read
 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.
PreClass Questions

Suppose T: ℝ^{4} → ℝ^{5} is a linear transformation and T(x) = Ax for some
matrix A. What are the dimensions of A? Why?

Give an example of a function T: ℝ^{2} → ℝ^{2} that is not a
linear transformation. Explain.

Give the standard matrix A for the linear transformation T: ℝ^{2} → ℝ^{2}
that rotates about the origin by an angle of π/3 radians counterclockwise.
 Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through Canvas
Week 4: Due Monday September 18 @ 11:59 pm
Multiplying and inverting matrices
To Read
 2.1 Matrix Operations
 2.2 Inverse of a Matrix
 2.3 Characterizations of Invertible Matrices
PreClass Questions

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, BA^{T}
 Let \( B = \begin{bmatrix} 1&0&1 \\ 0&1&2 \\0&2&5 \end{bmatrix} \). Find B^{1}

If A is an invertible matrix, are the rows of A a linearly independent set or a linearly dependent set?
Explain.
 Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through Canvas
Week 5: Due Monday September 25 @ 11:59 pm
Using matrices to shift, rotate, and skew graphics; The determinant function
To Read
 2.7 Applications to Computer Graphics
 3.1 Introduction to Determinants
 3.2 Properties of Determinants
PreClass Questions
 Why are homogenous coordinates used in computer graphics?
 Let \( A=\begin{bmatrix} 3&2\\4&1 \end{bmatrix} \). Find det(A).
 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
To Read
 4.1 Vector Spaces and Subspaces
 4.2 Null Spaces, Column Spaces, and Linear Transformations
PreClass Questions
 Give an example of a subset of ℝ^{2} that is a subspace of ℝ^{2}. Explain.

Give an example of a subset of ℝ^{2} that is not a subspace of
ℝ^{2}. Explain.
 If the columns of A are linearly independent, what is Nul(A)? Why?
 If A is 6 x 9 with 6 pivots, what is Col(A)? Why?
Submit answers through Canvas
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
To Read
 4.3 Linearly Independent Sets; Bases
PreClass Questions
Let \( A = \begin{bmatrix} 1&2&5 \\ 3&5&14 \\ 1&3&6 \end{bmatrix} \)
 Do the columns of A form a basis for ℝ^{3}? Explain.
 Give a basis for Col(A).
 Give a basis for Nul(A).
 Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through Canvas
Week 9: Due Monday October 23 @ 11:59 pm
Adjusting your reference; An invariant of vector spaces
To Read
 4.4 Coordinate Systems
 4.5 The Dimension of a Vector Space
 4.6 Rank
PreClass Questions

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 ℝ^{2}.
 Find \( [x]_{\mathcal{B}} \), the coordinates of \( x \) relative to the basis \( \mathcal{B} \).

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.
 Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through Canvas
Week 10: Due Monday October 30 @ 11:59 pm
Understanding longterm behavior; Directions fixed by matrix functions
To Read
 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.
PreClass Questions
 What is the point of studying Markov chains?
 What is the steady state vector for a stochastic matrix P?

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?
 Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through Canvas
Week 11: Due Monday November 6 @ 11:59 pm
Extending geometric intuition to higher dimensions
To Read
 6.1 Inner Product, Length, and Orthogonality
 6.2 Orthogonal Sets
PreClass Questions

Are the vectors \( u=\begin{bmatrix} 1\\2\\3 \end{bmatrix} \) and \( v=\begin{bmatrix} 2\\4\\2 \end{bmatrix}
\) orthongonal in ℝ^{3}? Explain.

Let H be the yzplane in ℝ^{3}.
 What is the orthogonal complement of H in ℝ^{3}?
 Give an orthogonal basis for H.

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 Canvas
Week 12: Due Monday November 13 @ 11:59 pm
Finding the closest vector
To Read
 6.3 Orthogonal Projections
 6.4 The GramSchmidt Process
 6.5 LeastSquares Problems
PreClass Questions

Let \( y=\begin{bmatrix} 1\\3\\5 \end{bmatrix} \) in ℝ^{3} and let W be the xyplane in
ℝ^{3}. Find the orthogonal projection of y onto W.
 What is the purpose of the GramSchmidt process?

Does every system Ax=b have a solution x? Does
every system Ax=b have a leastsquares solution \( \hat{x} \)? Explain.
Submit answers through Canvas
Week 13: Due Monday November 20 @ 11:59 pm
A factorization of certain square matrices
To Read
PreClass Questions
 What is the point of finding a diagonalization of a matrix?
 If A is 4 x 4 with eigenvalues 1, 2, 0, and 3, is A diagonalizable? Explain.
Submit answers through Canvas
Week 14: Due Monday November 27 @ 11:59 pm
A factorization of m x n matrices
To Read
 7.1 Diagonalization of Symmetric Matrices
 7.4 The Singular Value Decomposition
PreClass Questions
 Give an example of a 3x3 matrix A that is symmetric and a 3x3 matrix B that is not symmetric.

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?
 If A is any mxn matrix, what special property does the matrix A^{T}A have?
 Have you and your partner met to discuss the Problem Set? How much progress have you made?
Submit answers through Canvas
Week 15: Due Monday December 4 @ 11:59 pm
The power of the singular value decomposition
To Read
 When Life is Linear, Chapter 9, posted to Canvas
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.