Homework for MATH 5333 in Fall 2022 – Website of Dr. Michaela Vancliff

return to Fall 2022 MATH 5333 webpage

Required textbook:

Linear Algebra Done Right, 3rd Ed, by S. Axler, UTM, Springer.
A list of known errata for this book is posted here.
Videos by the author to accompany the book can be found here.

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Homework will not be collected.
There might be more than one correct answer for any given question.
Dates indicate homework assigned in lecture on that date. Dates for future assignments are tentative and subject to change.
If you notice some questions are in a nonincreasing order, then it means that the order is recommended by your instructor and is usually chosen to match the order in which the material was presented in class.
Any question of the form Hi (i = 1,….) is designed to help you answer some textbook questions – make sure you answer any Hi question before working on a textbook problem that is listed after it.
Any optional exercises are for those students who wish to explore some of the ideas in more depth – those questions are NOT game for the tests.
Skimming through the main ideas in a section shortly before that section is covered in class should help you understand the lecture – try it!
If any of the mathematics below does not display properly (e.g., an error message received), compare with this file.
Recall that our test dates are: Sept 14, Oct 12, Nov 9 and Dec 12.
LAST REVISION: 12/03/22.

Aug 22	Check your Canvas notifications to check you can receive Canvas announcements. Attendance will be noted, starting today. Read course syllabus carefully. Make a note of the test dates in your calendar. Review course website and repeat frequently during the semester. Review the course’s Canvas portal. Read this study tip and read https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively. Read your lecture notes (meaning the notes you should have taken during lecture) and & pgs 1-10, 12-17. Do 1A: 1, 2, 4, 11, 14 & 1B: 4. Read pgs 118-120.
Aug 24	Read your lecture notes & pgs 18-24. Do 1B: 6, 1, 2 & 1C: 1, 3-6, 10, 12, 14, 15. Do the following questions on past UTA preliminary examinations (see Canvas for how to access them): Jan 2013 #3(a), Jan 2014 #3, Jan 2015 #2(b). Read pgs 120-121.
Aug 29	Read your lecture notes & pgs 27-36. Do 1C: 20, 23 & 2A: 1-3, 5-7. Read pgs 121-122 and these notes about writing proofs. By this time, you have seen quite a few theorems. It is best NOT to memorize the theorems, but to do enough of the homework that the results of the theorems become known to you, though perhaps with different wording or using pictures.
Aug 31	Read your lecture notes & pgs 39-42, 44-47. Do 2A: 12-15 & 2B: 3-7 & 2C: 1-5, 8, 11-13, 15. Do the following questions on past UTA preliminary examinations: Jan 2013 #3(b)-(d), Aug 2014 #2(a), Jan 2015 #2(c)(d). Read pgs 122-123.
Sep 05	Labor Day Holiday (no lecture)
Sep 07	Read your lecture notes & pgs 51-57. Do 3A: 1, 3, 4, 7, 10, 11, 14. Read pgs 123-126. Test 1 will be 1 week from today; see Canvas for an information sheet.
Sep 12	Read your lecture notes & pgs 59-64. Do 3B: 3, 9-11, 29-31, 6, 13-15. Read pgs 65-66. Read pgs 126-129. Test 1 on Wednesday; see Canvas for an information sheet.
Sep 14	Test 1 today; see Canvas for an information sheet. Read lecture notes & pgs 70-77 & do 3C: 2, 12. Do the following questions on past UTA preliminary examinations: Jan 2013 #1(a), Jan 2014 #2, Aug 2014 #3(b)(c). Read pg 109.
Sep 19	Read your lecture notes and look over the solutions to Test 1 in the Assignments section of our Canvas portal. Read pgs 80-88 & do 3D: 1, 7(a), 8, 18.
Sep 21	Read your lecture notes & pgs 80-88 & do 3D: 15, 9-11, 20.
Sep 26	Application of dual spaces, dual maps and dual bases: if ${\color{white} A \in \mathbb F^{m\, , \, n}}$ , then dim(span of columns of ${\color{white} A}$ ) = dim(span of rows of ${\color{white} A}$ ). Proof dim(span of columns of ${\color{white} A}$ ) = dim(span( ${\color{white} \begin{array}{c}A e_1, \, A e_2, \ldots , A e_n \end{array}}$ )), where ${\color{white} \begin{array}{c}e_1, \, e_2, \ldots , e_n \end{array}}$ are the standard basis vectors in ${\color{white} \mathbb F^n}$ ${\color{white} = \dim(T(\mathbb F^n))}$ where ${\color{white} T : \mathbb F^n \to \mathbb F^m}$ is given by ${\color{white} T(v) = A v}$ for all ${\color{white} v \in \mathbb F^n}$ ${\color{white} = \dim(T'((\mathbb F^m)'))}$ = dim(span of columns of ${\color{white} A'}$ ) where ${\color{white} A'}$ is the matrix that represents ${\color{white} T'}$ = dim(span of columns of transpose of ${\color{white} A}$ ) = dim(span of rows of ${\color{white} A}$ ). Definition The rank of a matrix A is defined to be the dimension of the span of the columns of A. Read your lecture notes & pgs 101-112, and do question H1 at this website & 3F: 1, 3, 4, 7, 10, 12, 15, 18-22, 13, 34.
Sep 28	Read your lecture notes & pgs 131-136. Do 5A: 1-3, 8, 9, 11, 12, 14, 18-20, 25, 26, 32. Do the following questions on past UTA preliminary examinations: Aug 2013 #3(b)(ii).
Oct 03	Read your lecture notes & pgs 143-152. Do 5B: 1(a), 2, 4, 5 (hint: try p = x^m 1st), 7, 9, 13, 20.
Oct 05	Read your lecture notes & pgs 155-160. Do 5B: 14, 15, and 5C: 1-4, 6, 8, 9, 11, 14, 16 (hint for 16(d): write (0, 1) as a linear combination of the basis vectors found in (c) & then compute Tⁿ(0, 1) ). Do the following questions on past UTA preliminary examinations: Jan 2013 #2, Aug 2015 #1(a)(i). Test 2 will be 1 week from today; see Canvas for an information sheet. Look over Test 1 and its solutions posted on Canvas to study for Test 2. (Pay attention to how the solutions on the solution sheet are written; e.g., the level of detail provided in each solution.)
Oct 10	Read your lecture notes & pgs 163-170. Do 6A: 2, 4, 7, 19, 20, 24, 25, 26(a). Test 2 will be on Wednesday; see Canvas for an information sheet. Look over Test 1 and its solutions posted on Canvas to study for Test 2. (Pay attention to how the solutions on the solution sheet are written; e.g., the level of detail provided in each solution.)
Oct 12	Test 2 today; see Canvas for an information sheet. Look over Test 1 and its solutions posted on Canvas to study for Test 2. (Pay attention to how the solutions on the solution sheet are written; e.g., the level of detail provided in each solution.)
Oct 17	Read your lecture notes & pgs 170-174. Do 6A: 8, 10, 16. Read your lecture notes & pgs 180-186. Do 6B: 1(a), 3, 4.
Oct 19	Read your lecture notes & pgs 187-189. Do 6B: 5, 6,, 17, 7, 8. [Note that in problem 17, the map Ф is not explicitly defined, but can be inferred from the information provided in that problem.] Do the following questions on past UTA preliminary examinations: Jan 2015 #4(b), Aug 2015 #3 & #5(a)(b). Look over the solutions to Test 2 in the Assignments section of our Canvas portal.
Oct 24	Read your lecture notes & pgs 193-200. Do 6C: 1-3,, 5, 7. Do the following question on past UTA preliminary examinations: Aug 2014 #4(a).
Oct 26	Read your lecture notes & pgs 193-200 and do 6C: 11, 12. Do the following questions on past UTA preliminary examinations: Aug 2013 #5, Jan 2012 #3. The last theorem which we did not have time to prove in lecture today was THEOREM Suppose $\{ v_1, \ldots , v_n \}$ is a basis of a vector space $V.$ If $Q : V \to \mathbb F$ is a quadratic form on $V$ and if $b : V \times V \to \mathbb F$ is given by $Q$ as discussed in lecture, then the matrix $A = ( \ b(v_i, \, v_j)\ )$ is an $n \times n$ symmetric matrix and $b(v, \, w) = v^T A w$ for all $v, \, w \in V$ and $Q(v) = b(v, \, v) = v^T A v$ for all $v \in V.$ Conversely, if $A$ is an $n \times n$ symmetric matrix, then the formula $b(v, \, w) = v^T A w$ for all $v, \, w \in V$ defines a symmetric bilinear form $b : V \times V \to \mathbb F$ and $A_{ij} = b(v_i, \, v_j)$ for all $i, \, j.$ PROOF For any $n \times n$ matrix $C$ , we have $v_i^T C v_j$ is equal to the $ij$ -entry of $C$ . It follows that $A = ( \ b(v_i, \, v_j)\ )$ if and only if $b(v_i, \, v_j) = v_i^T A v_j$ for all $i, \, j,$ which holds if and only if $b(v, \, w) = b( \ \sum_{i = 1}^n \alpha_i v_i, \, \sum_{j = 1}^n \beta_j v_j \ ) = \sum_{i,\, j =1}^n \alpha_i \beta_j b(v_i, \, v_j) = \sum_{i,\, j =1}^n \alpha_i \beta_j v_i^T A v_j,$ which holds if and only if $b(v, \, w) = v^T A w$ for all $v, \, w \in V.$ (Here, $v = \sum_{i =1}^n \alpha_i v_i$ and $w = \sum_{j =1}^n \beta_j v_j$ , where $\alpha_i, \, \beta_j \in \mathbb F$ for all $i, \, j.$ ) The symmetry property follows since $b(w, \, v) = w^T A v = (v^T A^T w)^T = (v^T A w)^T$ for all $v, \, w \in V$ if and only if $A$ is symmetric, and since $(v^T A w)^T = b(v, \, w)^T = b(v, \, w)$ as $b(v, \, w)$ is a scalar. ♦ Do questions H2 & H3 at this website. Do the following question on past UTA preliminary examinations: Aug 2012 #4.
Oct 31	Read your lecture notes & pgs 203-212. Do 7A: 1, 2(hint: consider invertibility of T- λI), 3, 4, 6, 8, 9 and H4.
Nov 02	Read your lecture notes & pgs 212-214. Do 7A: 20, 21. Read your lecture notes & pgs 217-222. Do 7B: 6, 7, 9,, 11, H5, 2, 3, H6. Do the following questions on past UTA preliminary examinations: Jan 2014 #4 ( in (c), assume dim(V) finite ) & #5, Aug 2014 #4(b)(c), Jan 2015 #4(a), Aug 2015 #4, Jan 2015 #5, Aug 2015 #5(c). Read pgs 225-231. Test 3 will be 1 week from today; see Canvas for an information sheet. Look over Tests 1 & 2 and their solutions posted on Canvas to study for Test 3. (Pay attention to how the solutions on the solution sheets are written; e.g., the level of detail provided in each solution.)
Nov 07	Read your lecture notes & pgs 242-249. Do 8A: 1, 2, 4, 5, 16. Do the following questions on past UTA preliminary examinations: Aug 2015 #1(b)(i)(ii). Test 3 will be on Wednesday; see Canvas for an information sheet. Look over Tests 1 & 2 and their solutions posted on Canvas to study for Test 3. (Pay attention to how the solutions on the solution sheets are written; e.g., the level of detail provided in each solution.)
Nov 09	Test 3 today; see Canvas for an information sheet. Look over Tests 1 & 2 and their solutions posted on Canvas to study for Test 3. (Pay attention to how the solutions on the solution sheets are written; e.g., the level of detail provided in each solution.)
Nov 14	Read your lecture notes & pgs 242-249 & pgs 252-259. The last theorem which we did not have time to prove in lecture today was THEOREM Suppose $\mathbb F = \mathbb C$ and $T \in \mathcal L(V).$ If $\lambda_1, \ldots , \lambda_m$ are all the distinct eigenvalues of $T,$ with multiplicities $d_1, \ldots , d_m$ resp., then $V$ has a basis $\mathcal B$ such that $\mathcal M(T, \, \mathcal B) =\begin{bmatrix}A_1 & & & & \\ & A_2 & & \text{\huge0} & \\ & & \ddots & & \\ & \text{\huge0} & & A_{m-1} & \\ & & & & A_m\end{bmatrix},$ where each $A_j$ is a $d_j \times d_j$ matrix = $\left[\begin{array}{ccccc}\lambda_j &&&& \\ & \lambda_j & & \bigstar & \\ & & \ddots && \\ & \text{\huge0} & & \lambda_j & \\ & & & & \lambda_j\end{array}\right].$ PROOF We have $T\|_{G_{\lambda_j}} = (T - \lambda_j I)\|_{G_{\lambda_j}} + (\lambda_j I)\|_{G_{\lambda_j}}.$ Write $S_j = (T - \lambda_j I)\|_{G_{\lambda_j}}.$ By (c) of the Proposition in lecture, each $S_j$ is nilpotent on $G_{\lambda_j}.$ So there exists a basis $\mathcal B_j$ of $G_{\lambda_j}$ such that $\mathcal M (S_j, \, \mathcal B_j)$ is a strictly upper triangular $d_j \times d_j$ matrix. So $\mathcal M(T\|_{G_{\lambda_j}} , \, \mathcal B_j) = \mathcal M(S_j, \, \mathcal B_j) + \mathcal M((\lambda_j I)\|_{G_{\lambda_j}} , \, \mathcal B_j) =$ $\left[\begin{array}{ccccc}\lambda_j &&&& \\ & \lambda_j & & \bigstar & \\ & & \ddots && \\ & \text{\huge0} & & \lambda_j & \\ & & & & \lambda_j\end{array}\right]$ $= A_j.$ Let $\mathcal B = \bigcup_{j=1}^m \mathcal B_j =$ basis of V by (a) of Proposition. Hence, $\mathcal M(T, \, \mathcal B) =\begin{bmatrix}A_1 & & & & \\ & A_2 & & \text{\huge0} & \\ & & \ddots & & \\ & \text{\huge0} & & A_{m-1} & \\ & & & & A_m\end{bmatrix}.$ $\blacklozenge$ Do 8A: 6-9 and 8B: 1, 3(evals only), 5, 6. Do the following question on past UTA preliminary examinations: Aug 2015 #1(b)(iii).
Nov 16	Read your lecture notes & pgs 261-267. Do 8C: 1-3 & H 7-H9, 4-6, 12, 16, H10, H11. Look over the solutions to Test 3 in the Assignments section of our Canvas portal.
Nov 21	Read your lecture notes & pgs 261-267. Do the following questions on past UTA preliminary examinations: Jan 2013 #1(b) & Aug 2013 #1(c). Read your lecture notes & pgs 270-273.
Nov 23	No lectures nor office hours today; UTA offices open until noon. See the academic schedule at https://www.uta.edu/academics/academic-calendar
Nov 28	Read your lecture notes & pgs 270-273 and this file, which will be discussed in lecture on Wednesday. Do the following question on past UTA preliminary examinations: Jan 2015 #1(b)(c) & Aug 2014 #1(b). Do questions H12-H16
Nov 30	Make sure you understand the in-class handout on Jordan normal form. Do 8D: 4. Do the following question on past UTA preliminary examinations: Aug 2011 #3. Do questions H17-H19. Do the following questions on past UTA preliminary examinations: Jan 2016 #1(a) & Jan 2018 #1(a). (Optional: read pgs 91-93 & do 3E: 1, 6, 14(a).) Please remember to complete online the student feedback survey by 11 pm Tues Dec 6 — check your mymav e-mail for the link. I appreciate the feedback (e.g., quality of website, the choice of homework questions from the book and/or the questions from the additional homework, the information sheets and/or the solution sheets for the tests, the examples provided in class, the last handout with examples about Jordan Form, use of Canvas, quality of corny jokes,…..); thank you! I will have my usual office hours through Dec 5 inclusive, and I will also have additional office hours as noted below.
Dec 05	We will spend today’s lecture reviewing the course material & addressing students’ questions; no new homework. Please remember to complete online the student feedback survey by 11 pm Tues Dec 6 — check your mymav e-mail for the link. I appreciate the feedback (e.g., quality of website, the choice of homework questions from the book and/or the questions from the additional homework, the information sheets and/or the solution sheets for the tests, the examples provided in class, the last handout with examples about Jordan Form, use of Canvas, quality of corny jokes,…..); thank you! The Final Test will be on Monday Dec 12; see Canvas for an information sheet. I will have my usual office hours through today inclusive, and I will also have additional office hours as noted below.
Dec 07 1:15 PM	Office hour today 1:15-2:15 PM online only – see Canvas announcements for the Teams link. This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.
Dec 09 1:20 PM	Office hour today 1:20-2:20 PM in PKH 462 and online – see Canvas announcements for the Teams link. This is time the instructor is planning to be in PKH 462 and on Teams for students to “drop by” to ask questions.
Dec 10 2:45 PM	Office hour today 2:45-3:45 PM online only – see Canvas announcements for the Teams link. This is time the instructor is planning to be on Teams for students to “drop by” to ask questions.
Dec 12 5:30 PM	FINAL TEST today, starting at 5:30 PM; see Canvas for an information sheet. Look over Tests 1-3 and their solutions posted on Canvas to study for this test. (Pay attention to how the solutions on the solution sheets are written; e.g., the level of detail provided in each solution.)

The assignments from Fall 2019 can be viewed here .