Homework

Note: On each homework assignment, please write (i) your name, (ii) name of course, and (iii) Daily/Weekly Homework number.

Daily Homework

Here are the Daily Homework assignments for the semester. We will add to this list as we go. The Daily Homework assignments will be graded on a $\checkmark$-system. During class, you are only allowed to annotate your homework using the felt-tip pens that I will provide. I reserve the right to modify the assignment if the need arises.

Daily Homework 1: Complete the following tasks. (Due Wednesday, February 1)
1. Read Syllabus.
2. Read "How to use this book" in the Preface.
3. Read Section WILA (you can skim Subsection WILA.AA).
4. Complete exercises WILA.READ #1, 2
5. Write down at least three important things from the syllabus and/or the textbook.
Daily Homework 2: Complete the following tasks. (Due Friday, February 3)
1. Read Subsections SSLE.SLE and SSLE.PSS.
2. Complete exercises #1, 2 in SSLE.READ.
3. Complete exercises #C30-C34 in SSLE.EXC. Note 1: I'm leaving it to you to remind yourself how to solve systems of equations by substitution. If you are struggling, try reading the examples in Subsection SSLE.ESEO, and if that doesn't work, then ask me for help. Note 2: #C30-C34 are not in the physical textbook available on Lulu, but are in the PDF/HTML version available on the author's webpage.
4. Read Proof Technique D and Proof Technique T in Section PT of Appendix P.
Daily Homework 3: Complete the following tasks. (Due Monday, February 6)
1. Read Subsection SSLE.ESEO.
2. Complete exercises #C10, M40, M70, T20 in SSLE.EXC.
Daily Homework 4: Complete the following tasks. (Due Wednesday, February 8)
1. Read Subsections RREF.MVNSE and RREF.RO.
2. For each of exercises #C30-C34 in SSLE.EXC, identify the (i) coefficient matrix, (ii) constant vector, (iii) solution vector (with actual solutions), and (iv) augmented matrix. Note: Recall that these exercises are not in the physical textbook available on Lulu, but are in the PDF/HTML version available on the author's webpage.
3. Complete exercise #T10 in RREF.EXC.
Daily Homework 5: Complete the following tasks. (Due Friday, February 10)
1. Read Subsection RREF.RREF.
2. Complete exercises #C10, C13, C16, C19, C30, C31, C32 in RREF.EXC.
Daily Homework 6: Complete the following tasks. (Due Monday, February 13)
1. If necessary, re-read Subsection RREF.RREF.
2. Complete exercises #C12, C15, C18, C33 in RREF.EXC.
Daily Homework 7: Complete the following tasks. (Due Friday, February 17)
1. Read Section TSS.
2. Complete exercises #C22, C24, C25, C26, C28, M52, M57, T40 in TSS.EXC. For #C22-C28, use any method you'd like to row-reduce the matrix.
Daily Homework 8: Complete the following tasks. (Due Monday, February 20)
1. Read Section HSE.
2. Complete exercises #C30, C31, M50, M51, M52, T10, T20 in HSE.EXC.
Daily Homework 9: Complete the following tasks. (Due Wednesday, February 22)
1. Read Section NM.
2. Complete exercises #C30, C31, C32, M51, M52 in NM.EXC.
Daily Homework 10: Complete the following tasks. (Due Friday, February 24)
1. Read Section VO.
2. Complete exercise #3 in VO.READ.
3. Complete exercises #C10, T13, T17, T18 in VO.EXC.
Daily Homework 11: Complete the following tasks. (Due Monday, February 27)
1. Read Section LC.
2. Complete exercises #2, 3 in LC.READ.
3. Complete exercises #C40, C41, M10 in LC.EXC.
Daily Homework 12: Complete the following tasks. (Due Monday, March 5)
1. Read Section SS.
2. Complete exercises #C40, C41, C42, M10, M11 in SS.EXC.
Daily Homework 13: Complete the following tasks. (Due Wednesday, March 7)
1. Read Section LI.
2. Complete exercise #C50 in SS.EXC.
3. Complete exercises #C21, C22, C31, M21, M51 in LI.EXC.
Daily Homework 14: Complete the following tasks. (Due Friday, March 9)
1. Read Section LDS.
2. Complete exercises #1, 2 in LDS.READ.
3. Complete exercises #C50, C51 in LDS.EXC.
Daily Homework 15: Complete the following tasks. (Due Monday, March 12)
1. Let $\vec{u}=\begin{bmatrix}1\\ 1\\-1\\2 \end{bmatrix}$ and $\vec{v}=\begin{bmatrix}2\\ 1\\ 3 \\0 \end{bmatrix}$.
  - Find $||\vec{u}||, ||\vec{v}||, ||\vec{u}+\vec{v}||$. Is $||\vec{u}+\vec{v}||$ equal to $||\vec{u}||+||\vec{v}||$?
  - Find $\langle \vec{u}, \vec{v}\rangle$. Are $\vec{u}$ and $\vec{v}$ orthogonal?
2. Let $A$ be a coefficient matrix and let $\vec{x}$ be a solution to $\mathcal{LS}(A,\vec{0})$. Explain why $\vec{x}$ is orthogonal to the rows (as vectors) of $A$.
3. Let \[S=\left\{ \begin{bmatrix}4\\ -1\\1 \end{bmatrix}, \begin{bmatrix}-1\\ 0\\4 \end{bmatrix}, \begin{bmatrix}-4\\ -17\\-1 \end{bmatrix}\right\}.\]
  - Determine whether $S$ is an orthogonal set.
  - Using your previous answer, determine whether $S$ is linearly independent.
4. Let \[T=\left\{ \begin{bmatrix}4\\ -3\\0 \end{bmatrix}, \begin{bmatrix}1\\ 2\\0 \end{bmatrix}, \begin{bmatrix}0\\ 0\\4 \end{bmatrix}\right\}.\]
  - Determine whether $T$ is linear independent.
  - Is $T$ an orthogonal set?
  - What does this example illustrate?
5. Complete exercise #T20 (use $\mathbb{R}^m$ and $\mathbb{R}$ instead of $\mathbb{C}^m$ and $\mathbb{C}$) in O.EXC.
Daily Homework 16: Complete the following tasks. (Due Wednesday, March 14)
1. Let $\vec{u}=\begin{bmatrix}0\\ 1\\3 \\-6 \end{bmatrix}$, $\vec{v}=\begin{bmatrix}-1\\ 1\\ 2\\ 2 \end{bmatrix}$. Find $\mathrm{proj}_{\vec{u}}\vec{v}$ and $\mathrm{proj}_{\vec{v}}\vec{u}$.
2. Let $\vec{u}, \vec{v}\in\mathbb{R}^m$. Suppose that $\vec{u}$ and $\vec{v}$ are orthogonal. What can you say about $\mathrm{proj}_{\vec{u}}\vec{v}$ and $\mathrm{proj}_{\vec{v}}\vec{u}$?
Daily Homework 17: Complete the following tasks. (Due Friday, March 16)
1. Let \[S=\left\{ \begin{bmatrix}4\\ -1\\1 \end{bmatrix}, \begin{bmatrix}-1\\ 0\\4 \end{bmatrix}, \begin{bmatrix}-4\\ -17\\-1 \end{bmatrix}\right\}.\] (See Daily Homework 15.) Determine whether $S$ is orthonormal. Explain your answer. Create an orthonormal set $T$ such that $\langle T\rangle =\langle S\rangle$.
2. Let \[S=\left\{ \begin{bmatrix}3\\ 4\\0\\0 \end{bmatrix}, \begin{bmatrix}-1\\ 1\\0\\0 \end{bmatrix}, \begin{bmatrix}2\\ 1\\ 0\\ -1 \end{bmatrix}, \begin{bmatrix}0\\ 1\\ 1\\ 0 \end{bmatrix}\right\}.\] Use the Gram-Schmidt process to find an orthonormal set $T$ such that $\langle T\rangle =\langle S\rangle$.
3. Let $S=\{\vec{v}_1,\ldots,\vec{v}_p\}\subseteq \mathbb{R}^m$ be a linearly independent set. Let $T=\{\vec{u}_1,\ldots,\vec{u}_p\}$ be the set that results from doing Gram-Schmidt on $S$. Prove that $\vec{u}_1$ is orthogonal to an arbitrary $\vec{u}_i$ with $i\neq 1$. Note: This exercise is asking you to prove a special case of the proof of the Gram-Schmidt process. Also, you may assume that $\vec{u}_1$ is orthogonal to $\vec{u}_2,\vec{u}_3,\ldots,\vec{u}_{i-1}$.
4. Read Section MO.
5. Complete exercise #1 in MO.READ.
6. Complete exercise #T18 in MO.EXC.
Daily Homework 18: Complete the following tasks. (Due Monday, March 26 or Tuesday, March 27)
1. Complete exercise #3 in MO.READ.
2. Complete exercises #1,3 in MM.READ.
Daily Homework 19: Complete the following tasks. (Due Wednesday, March 28)
1. Read Section MM.
2. Complete exercise #2 in MM.READ.
3. Complete exercises #C20 (do by hand, not with computer or calculator; you don't need to do it twice), T31, T32, T50 in MM.EXC.
Daily Homework 20: Complete the following tasks. (Due Friday, March 30)
1. Read Sections MISLE and MINM.
2. Complete exercises #C21, C28, C40, T10 in MISLE.EXC.
3. Complete exercise #C40 in MINM.EXC.
Daily Homework 21: Complete the following tasks. (Due Wednesday, April 4)
1. Read Section CRS.
2. Complete exercises #1, 2, 3 in CRS.READ.
3. Complete exercises #C31, C33, M20, T40 in CRS.EXC.
Daily Homework 22: Complete the following tasks. (Due Friday, April 6)
1. Read Section VS.
2. Complete exercises #M11, M12, M13, M15, M20, M21 in VS.EXC.
Daily Homework 23: Complete the following tasks. (Due Monday, April 9)
1. Read Section S.
2. Complete exercise #2 in S.READ.
3. Complete exercises #C25, C26, M20, T30, T31 in S.EXC.
Daily Homework 24: Complete the following tasks. (Due Wednesday, April 11)
1. Re-read Section S.
2. Complete exercises #C15, C16, C20, C21 in S.EXC.
Daily Homework 25: Complete the following tasks. (Due Friday, April 13)
1. Read Section LISS.
2. Complete exercises #C20, C21, C22, C40 in LISS.EXC.
Daily Homework 26: Complete the following tasks. (Due Monday, April 16)
1. Complete exercises #C41, C42 in LISS.EXC.
2. Complete exercises #C10, C12, C13 in B.EXC.
Daily Homework 27: Complete the following tasks. (Due Monday, April 23)
1. Read Section D.
2. Complete exercises #C21, C31, C35, C36, M20, M21 in D.EXC.
Daily Homework 28: Complete the following tasks. (Due Wednesday, April 25)
1. Skim over Section PD and Chapter D.
2. Complete exercise #3 in PD.READ.
3. Complete exercises #C40, T15 in PD.EXC.
4. Complete exercises #C24, C26 in DM.EXC.
Daily Homework 29: Complete the following tasks. (Due Friday, April 27)
1. Read Section EE.
2. For each of the following matrices, determine the eigenvalues and then geometrically describe the corresponding sets of eigenvectors. Hint: one of these is a rotation, one of these is a projection, and one of these just stretches.
  - $A=\begin{bmatrix}2 & 0\\0 & 2 \end{bmatrix}$
  - $B=\begin{bmatrix}1 & 0\\0 & 0 \end{bmatrix}$
  - $C=\begin{bmatrix}0 & -1\\1 & 0 \end{bmatrix}$
3. Complete exercise #T10 in EE.EXC.
Daily Homework 30: Complete the following tasks. (Due Monday, April 30)
1. Re-read Section EE.
2. Complete exercises #C19, C21, C23, C26 in EE.EXC.
Daily Homework 31: Complete the following tasks. (Due Wednesday, May 2)
1. Read Section PEE.
2. Complete exercises #T10, T20 in PEE.EXC.
3. Let $A$ be an $n\times n$ matrix. Prove that if all the values off the diagonal (from upper left to lower right) of $A$ are 0, then the eigenvalues of $A$ are precisely the entries on the diagonal.
Daily Homework 32: Complete the following tasks. (Due Friday, May 4)
1. Read Section LT.
2. Complete exercises #C25, C26, C40 (assume $T$ is linear) in LT.EXC.
3. Define $T:\mathbb{R}^2\to \mathbb{R}^2$ via $T\left(\begin{bmatrix}x\\y \end{bmatrix}\right)=\begin{bmatrix}x^2\\y \end{bmatrix}$. Determine whether $T$ is linear or not. Justify your answer.
Daily Homework 33: Complete the following tasks. (Due Monday, May 7)
1. Complete exercises #1, 2 in LT.READ.
2. Complete exercises #C16, C41 in LT.EXC.
Daily Homework 34: Complete the following tasks. (Due Wednesday, May 7)
1. Complete exercises #C20, C25, C40 in ILT.EXC.
2. Complete exercise #2 in SLT.READ.

Weekly Homework

In addition to the Daily Homework, you will also be required to submit formally written solutions to selected problems each week. Assignments will be added as we go.

Weekly Homework 1: Complete the following tasks. (Due Tuesday, February 7 by 5PM)
1. See the forum post titled "PSU Sage usernames and passwords". Follow the instructions for how to change your password.
2. Go to the "Home" directory of your Sage account. Click on the menu item "Published" located at the top right. This will take you to a list of published worksheets.
3. Find the worksheet titled "Linear Algebra - Weekly Homework 1" and click on it.
4. This will take you to a static version of the worksheet, but we want you to be able to edit it. Click on "Edit this" in the upper left corner. This will open your very own copy of the worksheet for you to play with. You can also find the published worksheet here.
5. After you've got the worksheet open, just follow along.
6. If you are having trouble, just let me know.
Weekly Homework 2: Complete the following tasks. (Due Tuesday, February 14 by 5PM)
1. Determine the solution set for the following systems of equations using any method that you'd like. Also, describe in words what the solution set looks like. \[\begin{align} x_1-2x_2+x_3= & 4\\ 3x_1-6x_2+4x_3 = & 16\\ 2x_1-4x_2+3x_3 = & 12\\ \end{align}\]
2. Determine the solution set for the following system of equations by doing row operations by hand. Also, describe in words what the solution set looks like. \[\begin{align} x_1 + 3x_2 + x_3 = & 1 \\ -4x_1 -9x_2 + 2x_3 = & -1 \\ -3x_2 -5x_3 = & -4 \end{align}\]
3. If possible, provide an example of each of the following. If no example exists, explain why. Try to make your examples as simple as possible. In particular, feel free to use only 0's and 1's to fill your matrices.
  - A matrix in RREF that corresponds to a system of linear equations with 4 equations in 3 variables having a unique solution.
  - A matrix in RREF that corresponds to a system of linear equations with 3 equations in 4 variables having 2 pivots and no solution.
  - A matrix in RREF that corresponds to a system of linear equations with 3 equations in 3 variables having no solution.
  - A matrix in RREF that corresponds to a system of linear equations with 4 equations in 4 variables having 2 pivots and infinitely many solutions.
  - A matrix in RREF that corresponds to a system of linear equations with 3 equations in 3 variables having 2 pivots and a unique solution.
  - A matrix in RREF that corresponds to a system of linear equations with 4 equations in 3 variables having 3 pivots and infinitely many solution.
Weekly Homework 3: Complete the following tasks. (Due Tuesday, February 21 by 5PM)
1. Consider the following system of equations. \[\begin{align} 2x_1+x_2+7x_3−7x_4 = & 8\\ −3x_1+4x_2−5x_3−6x_4 = & -12\\ x_1+x_2+4x_3−5x_4 = & 4 \end{align}\]
  - Determine the solution set by doing row operations by hand.
  - It turns out that this system has infinitely many solutions. List 3 of them.
  - Describe the solution set geometrically.
2. Suppose $\mathcal{LS}(A,\vec{b})$ is a linear system. Describe what happens in each of the following scenarios. Be as specific as possible and justify your answer.
  - The $i$th row of $A$ is identical to the $j$th row.
  - The $i$th row of $A$ is a nonzero multiple of the $j$th row.
  - The $k$th row of $A$ is the sum of the $i$th row and the $j$th row.
  - The $k$th row of $A$ is the sum of a nonzero multiple of the $i$th row and a nonzero multiple (not necessarily the same multiple) of the $j$th row.
3. Suppose $\mathcal{LS}(A,\vec{b})$ is a linear system with 3 equations in 3 variables. Describe all possibilities for pivots and free variables for this system. In each case, state whether the system has no solution, a unique solution, or infinitely many solutions.
Weekly Homework 4: Complete the following tasks. (Due Friday, March 2 by 5PM)
1. Consider each of the following claims. If the claim is true, briefly justify it. If it is false, provide a counterexample.
  - If $\mathcal{LS}(A,\vec{b})$ is consistent, then $\mathcal{LS}(A,\vec{0})$ is consistent.
  - If $\mathcal{LS}(A,\vec{0})$ is consistent, then $\mathcal{LS}(A,\vec{b})$ is consistent.
  - The system $\mathcal{LS}(A,\vec{b})$ has a unique solution iff $\mathcal{LS}(A,\vec{0})$ has a unique solution.
2. Find the null space of the following matrix and then state whether the matrix is singular, nonsingular, or neither. \[\begin{bmatrix} 2 & 1 & -1 & -9\\ 1 & 2 & -8 & 0\\ 2 & 2 & -6 & -6\\ -1 & 2 & -12 & -12 \end{bmatrix}\]
3. Find $\alpha$ and $\beta$ that solve the following vector equation. \[\alpha\begin{bmatrix}1\\ 1 \end{bmatrix}+\beta \begin{bmatrix}-1\\ 2 \end{bmatrix}=\begin{bmatrix}5\\ -1 \end{bmatrix}\]
Weekly Homework 5: Complete the following tasks. (Due Thursday, March 14 by 5PM)
1. Let $\vec{u}=\begin{bmatrix}1\\2\\3 \end{bmatrix}$ and $\vec{v}=\begin{bmatrix}4\\ 5\\ 6 \end{bmatrix}$. Find a vector $\vec{w}$ such that $\langle \{\vec{u}, \vec{v} \}\rangle =\langle \{\vec{u}, \vec{v}, \vec{w}\}\rangle$.
2. Let \[S=\left\{\begin{bmatrix}4\\4\\7 \end{bmatrix}, \begin{bmatrix}-5\\-4\\-6 \end{bmatrix}, \begin{bmatrix}-2\\-1\\-1 \end{bmatrix}, \begin{bmatrix}1\\0\\-1 \end{bmatrix}, \begin{bmatrix}3\\3\\6 \end{bmatrix}\right\}.\] Find a linearly independent subset $T$ of $S$ such that $\langle S\rangle =\langle T\rangle$.
3. Let \[A=\begin{bmatrix}1 & -1 & -2 & 5 & 2\\ 1 & 1 & 2 & 5 & 1\\ 2 & 3 & 6 & 7 &1\\ 1 & 2 & 4 & 2 & 0\end{bmatrix}.\] Find linearly independent set of vectors $S$ such that $\mathcal{N}(A)=\langle S\rangle$.
4. Let $S\subseteq \mathbb{R}^m$ such that $\vec{0}\in S$. Prove that $S$ is linearly dependent.
5. If possible provide a simple example of a $4\times 5$ matrix $A$ such that (i) the first, second, and fourth columns of $A$ form a linearly independent set, and (ii) $\mathcal{N}(A)$ is a plane in $\mathbb{R}^5$. If this is not possible, explain why.
Weekly Homework 6: Complete the following tasks. (Due Thursday, April 5 by 5PM)
1. Let $A$ and $B$ be given by the following matrices. \[\ A=\begin{bmatrix}4 & 1 & 2\\ 1 & 0 & 1\\ 3 & 1 & 5\end{bmatrix}, B= \begin{bmatrix}1 & 3 & 5\\ -1 & 2 & 1\\ 0 & 1 & 0\end{bmatrix} \] Compute $AB$ by hand.
2. Let \[ A=\begin{bmatrix}2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\end{bmatrix}. \] Compute $A^2$, $A^3$, and $A^4$. Then conjecture a formula for $A^n$ for any $n\in \mathbb{N}$. You are welcome to use Sage or your calculator, but be sure to explain what you are doing.
3. For each of the following matrices, doing the computations by hand, find the inverse if it exists. If the inverse does not exist, explain why.
  - $A=\begin{bmatrix}1 & 0 & 1\\ 2 & -1 & 1\\ 1 & 1 & 1\end{bmatrix}$
  - $B=\begin{bmatrix}2 & -1 & 1\\ 3 & 1 & 2\\ 1 & 2 & 1\end{bmatrix}$
4. Use an inverse matrix to solve the following system of equations. You are welcome to use Sage or your calculator, but be sure to explain what you are doing. \[\begin{align} x_1+2x_2-x_3 = & 1\\ −x_1-4x_2 = & 2\\ 2x_1+5x_2-x_3 = & 3 \end{align}\]
5. Suppose $A$ is an $n\times n$ matrix such that $A$ is singular. What does Theorem NME3 tell us? Be as explicit as possible.
Weekly Homework 7: Complete the following tasks. (Due Thursday, May 3 by 5PM)
1. Let \[A=\begin{bmatrix}1 & -2 & 1 & 1\\ 2 & -4 & -1 & 3\\ 1 & -2 & 0 & 0\\ -2 & 4 & 1 & 0\end{bmatrix}\] Using the above matrix, answer the following questions.
  - Find the rank of $A$.
  - Find the nullity of $A$.
  - Explain the geometric significance of the values for the rank and nullity.
  - What is the determinant of $A$? Based on your previous answers, you should know the answer without having to compute it directly.
2. Let \[B=\begin{bmatrix}-2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0\end{bmatrix}\] Using the above matrix, answer the following questions.
  - Find the eigenvalues for $B$. You should be able to do this very quickly! But be sure to justify your answer.
  - For each eigenvalue, find a basis for the corresponding eigenspace and then describe it geometrically.
  - The matrix $B$ determines a function from $\mathbb{R}^3$ to $\mathbb{R}^3$. Describe this function geometrically.
3. Let \[C=\begin{bmatrix}-1 & 2 & 2\\ 2 & 2 & -1\\ 2 & -1 & 2\end{bmatrix}\] Using the above matrix, find the eigenvalues for $C$ and then determine the dimension for each of the corresponding eigenspaces. Compute the eigenvalues by hand. You can use Sage or your calculator to help you find the dimensions, but justify your answers.

Getting Help

There are many resources available to get help. First, I recommend that you work on homework in groups as much as possible, and to come see me whenever necessary. Also, you are strongly encouraged to ask questions in the course forum on our Moodle page page, as I will post comments there for all to benefit from. To effectively post to the course forum, you will need to learn the basics of LaTeX, the standard language for typesetting in the mathematics community. See the Quick LaTeX guide for help with $\LaTeX$. If you need additional help with $\LaTeX$, post a question in the course forum on our Moodle page.

You can also visit the Math Activity Center, which is located in Hyde 351. This student-run organization provides peer tutoring services for most 1000 and 2000 level math courses and some 3000 level courses. Tutors are typically math majors interested in teaching math and practicing their instructional skills. You can drop in anytime during open hours.

Lastly, you can always .