Homework for MATH 5333 in Fall 2025 – Website of Dr. Michaela Vancliff

return to Fall 2025 MATH 5333 webpage

Required (free) textbook:

Linear Algebra Done Right, 4th 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 .
Any UTA students who are vision impaired and need help reading the mathematics on this webpage should consult with UTA’s SARC for assistance.
The homework from Fall 2022 can be viewed here . However, that homework was assigned using the previous edition of the textbook, so the page numbers and question numbers will differ from those of the new edition.
Recall that our test dates are: Sept 11, Oct 9, Nov 6 and Dec 9.
LAST REVISION: 11/23/25.

Aug 19	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 list of study techniques 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, 12-16. Do 1A: 1, 7, 9 10 13 & 1B: 4. Read pgs 119-120.
Aug 21	Read your lecture notes & pgs 18-24. Do 1B: 6, 8, 1, 2 & 1C: 1, 3-6, 10, 12, 15, 14. 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 pg 121.
Aug 26	Read your lecture notes & pgs 27-36. Do 1C: 20, 21, 23 & 2A: 2-8, 11, 13. Read pgs 122-123 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 28	Read your lecture notes & pgs 39-42, 44-48. Do 2A: 15-18 & 2B: 3, 4, 6, 7. Read pgs 123-124.
Sep 02	Read your lecture notes & pgs 44-48. Do 2C: 1-5, 7, 10-14 and the following questions on past UTA preliminary examinations: Jan 2013 #3(b)-(d), Aug 2014 #2(a), Jan 2015 #2(c)(d). Read pgs 124-125.
Sep 04	Read your lecture notes & pgs 52-56. Do 3A: 1, 3, 4, 7, 12, 13, 16. Read pg 126. Test 1 will be 1 week from today; check Canvas for an information sheet.
Sep 09	Read your lecture notes & pgs 59-65. Do 3B: 1, 2, 6, 9, 10, 12-14, 27, 30. Read pg 127. Test 1 will be on Thursday; check Canvas for an information sheet.
Sep 11	Read your lecture notes & pgs 69-73. Do 3C: 4 and the following questions on past UTA preliminary examinations: Jan 2013 #1(a), Jan 2025 #3(a)(b), Aug 2025 #3(b)(i)(ii). Read pgs 127-128. Test 1 is today; check Canvas for an information sheet.
Sep 16	Read your lecture notes & pgs 69-73. Do 3C: 6, 7, 14. Read pgs 128-129. Look over the solutions to Test 1 on Canvas; use password “math5333” to open the file. Read through the suggestions of study techniques from Aug 19 above and see which one(s) might work for you.
Sep 18	Read your lecture notes & pgs 76-79 & pgs 82-84. Do 3C: 17 and 3D: 1, 4, 9, 10 (hint: use FTLM as suggested by book’s hint or consider matrices).
Sep 23	Read your lecture notes & pgs 84-93. Do 3D: 20, 21, 23, 24.
Sep 25	Read your lecture notes & pgs 105-107. Do 3F: 1, 3, 4, H1, 9, 10 (hint: use #9), 11, where H1 is H1: Suppose, without proof, that B = $\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\;\begin{bmatrix}1\\1\\0\end{bmatrix},\;\begin{bmatrix}1\\1\\1\end{bmatrix}\right\}$ is a basis of ℝ³. Find the dual basis of the dual space ( ℝ³ )′ relative to B.
Sep 30	Read your lecture notes & pgs 107-113. Do 3F: 13-15, 22 (hint: consider dimension for (b)).
Oct 02	Read your lecture notes & pgs 135-139. [Optional: read pgs 132-134.] Do 5A: 1, 8-10, 13, 19-21, 25, 26 (hint: use #25). Do the following question on past UTA preliminary examinations: Aug 2013 #3(b)(ii). Test 2 will be 1 week from today; check Canvas for an information sheet.
Oct 07	Read your lecture notes & pgs 143-144, top 1/2 of pg 145 & pgs 148-150, especially 5.29, 5.31 and 5.32. Do 5B: 1, 3, 11, 12, H2, 14, 20, 24, 22, where H2 is H2: (a) Find the minimal polynomial (in factored form) of each of the following matrices: (i) $\begin{bmatrix}2&0\\0&2\end{bmatrix},$ (ii) $\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix},$ (iii) $\begin{bmatrix}2&1\\0&2\end{bmatrix},$ (iv) $\begin{bmatrix}2&1&0\\0&2&0\\0&0&2\end{bmatrix},$ (v) $\begin{bmatrix}2&1&0\\0&2&1\\0&0&2\end{bmatrix},$ (vi) $\begin{bmatrix}0&1\\0&0\end{bmatrix},$ (vii) $\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix},$ (viii) $\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix},$ (ix) $\begin{bmatrix}2&0\\0&3\end{bmatrix},$ (x) $\begin{bmatrix}2&0&0\\0&2&0\\0&0&3\end{bmatrix},$ (xi) $\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}.$ (b) For each matrix in (a), describe the factored minimal polynomial in terms of the appearance of the matrix. What is the pattern? Test 2 will be on Thursday; check Canvas for an information sheet.
Oct 09	Test 2 is today; check Canvas for an information sheet.
Oct 14	Read your lecture notes & pgs 154-160 and do 5C: 1, 4, 5, 8, 11. Do the following questions on past UTA preliminary examinations: Jan 2013 #2, Aug 2015 #1(a)(i). Look over the solutions to Test 2 on Canvas; use password “math5333” to open the file. Read through the suggestions of study techniques from Aug 19 above and see which one(s) might work for you.
Oct 16	Read your lecture notes & pgs 163-170 and do 5D: 1, 3, 6, 9, 11, 14 (hint: for (b), consider min polys), and read Theorems 5.75 & 5.76 and their proofs.
Oct 21	Read your lecture notes & pgs 181-191 and do 6A: 2, 3(b), 26, 4, 9, 11, 21 (& read the poem on pg 196 ).
Oct 23	Read your lecture notes & pgs 197-206 and do 6B: 7, 8, 11, 12, 17, 18. Do the following questions on past UTA preliminary examinations: Jan 2015 #4(b), Aug 2015 #3. Read pgs 211-212 and do 6C: 1 and the following questions on past UTA preliminary examinations: Jan 2012 #3, Jan 2020 #4.
Oct 28	Read your lecture notes & pgs 212-219 and pgs 228-229 and do 6C: 15, 17 and 7A: 1 and the following question from past UTA preliminary examinations: Jan 2020 #5(a).
Oct 30	Read your lecture notes & pgs 230-238 and do 7A: 3 (hint: consider invertibility of T – λ I), 22, 23, 30, 31, H3, where H3 is H3: Let $\mathbb F=\mathbb R$ and $V=\mathbb{R}\begin{bmatrix}1\\1\end{bmatrix}\subset\mathbb{R}^2=W$ with dot product for the inner product. Define $T\in\mathcal{L}(V,\,W)$ by $T(v)=\begin{bmatrix}1&2\\3&4\end{bmatrix}v$ for all $v\in V.$ In particular, $T\left(\begin{bmatrix}a\\a\end{bmatrix}\right)=\begin{bmatrix}3a\\7a\end{bmatrix}$ for all $a\in\mathbb{R}.$ (a) You may assume, without proof, that the adjoint, $T^{}$ of $T$ is given by $T^\left(\begin{bmatrix}1\\0\end{bmatrix}\right)=\frac32\begin{bmatrix}1\\1\end{bmatrix}$ and $T^\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\frac72\begin{bmatrix}1\\1\end{bmatrix}.$ (∗) Compute $\langle T(v),\,w\rangle$ and $\langle v,\,T^(w)\rangle$ for all $v=\begin{bmatrix}a\\a\end{bmatrix}\in V$ and $w=\begin{bmatrix}b\\c\end{bmatrix}\in W$ (where $a,\,b,\,c\in\mathbb R),$ and verify that they are equal for all $a,\,b,\,c\in\mathbb R.$ (b) For $T,\,V,\,W$ as in (a), prove the given assumption (∗). (Hint: use orthonormal bases.) Test 3 will be 1 week from today; check Canvas for an information sheet.
Nov 04	Read your lecture notes & pgs 243-247 and do 7B: 11, 21, 1-6 and the following questions from past UTA preliminary examinations: Jan 2015 #4(a), 5. Test 3 will be on Thursday; check Canvas for an information sheet.
Nov 06	Test 3 is today; check Canvas for an information sheet.
Nov 11	Read your lecture notes & pgs 298-303 and do 8A: 1, 2, 6, 10. Look over the solutions to Test 3 on Canvas; use password “math5333” to open the file. Read through the suggestions of study techniques from Aug 19 above and see which one(s) might work for you.
Nov 13	Read your lecture notes & pgs 303-305 & pgs 308-315. Do 8A: 13, 17(a), 22 and the following question from past UTA preliminary examinations: Aug 2020 #1(b) and 8B: 3, 4, 8, 11-16, 20 and the following question from past UTA preliminary examinations: Aug 2015 #1(b).
Nov 18	Read your lecture notes & pgs 321-324 and the last example on the in-class handout on Jordan Normal (Canonical) Form and do 8C: 2(a)(hint: find all eigenvalues), 6, 10 and do H4. Suppose 𝔽 = ℂ. Find the Jordan normal (canonical) form J of the matrix A = $\begin{bmatrix}1&2\\0&3\end{bmatrix}$ and find an invertible matrix P such that P^-1 A P = J. and do the following question from past UTA preliminary examinations: Jan 2020 #1(b) and do H5. Let T ∈ 𝓛(V) and let 𝓑₁ and 𝓑₂ denote two bases of V. (a) Find an example with V = 𝔽² where 𝓜(T, 𝓑₁ , 𝓑₁) and 𝓜(T, 𝓑₁ , 𝓑₂) do not have the same eigenvalues. (b) Why does (a) not contradict the fact that similar matrices have the same eigenvalues? (See previous question for definition of “similar”.) and do the following question from past UTA preliminary examinations: Aug 2013 #1(c) and do H6. Suppose 𝔽 = ℝ. Find the Jordan normal (canonical) form J of the matrix A = $\begin{bmatrix}37&10&-5&5\\10&22&10&-10\\-5&10&37&5\\5&-10&5&37\end{bmatrix}$ , given that the characteristic polynomial of A is (x – 7)(x – 42)³ , and find an *orthogonal* matrix P such that P^t A P = J, where P^t denotes the transpose of the matrix P. and do the following questions from past UTA preliminary examinations: Aug 2020 #1(c) & Aug 2025 #1(b).
Nov 20	Read your lecture notes (if any) and the in-class handout on Jordan Normal (Canonical) Form and do 8C: 5 and do H7. Suppose 𝔽 = ℂ. The matrix A = $\begin{bmatrix}7&-6&-3\\8&-7&-4\\-4&4&3\end{bmatrix}$ has characteristic polynomial (x – 1)³ . (a) How many linearly independent eigenvectors does A have? Is this information, together with the characteristic polynomial, sufficient for determining the Jordan normal (canonical) form J of A (up to rearrangement of blocks)? Explain your answer. If this information is enough, what is J? (b) Find a matrix P such that P^-1 A P = J. and do the following questions from past UTA preliminary examinations: Aug 2019 #1(a) & Jan 2016 #1(a) & do H8. Suppose 𝔽 = ℂ. Find the Jordan normal (canonical) form J of the matrix A = $\begin{bmatrix}4&1&-1&-1\\1&4&-1&-1\\1&3&0&-3\\1&-1&1&4\end{bmatrix}$ given that its characteristic polynomial is (x − 3)⁴, and find an invertible matrix P such that P^-1 A P = J. and do the following question from past UTA preliminary examinations: Aug 2025 #1(a). Please remember to complete online the student feedback survey by 11:59 pm on Dec 2 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the test information sheets and/or the solution sheets for the tests, the examples provided in class, the JNF handout, corniness of jokes …??). Thank you!
Nov 25	Read your lecture notes & pgs 333-334 and 337 and do 9A: 10 and H9. Let V denote a vector space over 𝔽 and let b denote a bilinear form on V. For u, v ∈ V and α, β ∈ 𝔽, compute b( α u + β v, α u + β v ). H10. For each of the following symmetric matrices, give the corresponding quadratic form: (a) $\begin{bmatrix}1&2&3\\2&4&5\\3&5&6\end{bmatrix}$ (b) $\begin{bmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}$ (c) $\begin{bmatrix}1&0&-1&0\\0&3&0&2\\-1&0&0&0\\0&2&0&2\end{bmatrix}$ . H11. For each of the following quadratic forms Q, find a symmetric matrix A such that Q(v) = v^tAv for all v ∈ V (here, v^t denotes the transpose of v): (a) Q(x₁ , x₂) = 6 x₁² – 7 x₁ x₂ + 8 x₂² for all x₁ , x₂ ∈ 𝔽 (b) Q(x₁ , x₂) = x₁ x₂ for all x₁ , x₂ ∈ 𝔽 (c) Q(x₁ , x₂ , x₃) = x₁ x₂ for all x₁ , x₂ , x₃ ∈ 𝔽 (d) Q(x₁ , x₂ , x₃) = 3 x₁² + 4 x₂² + 5 x₃² + 6 x₁ x₃ + 7 x₂x₃ for all x₁ , x₂ , x₃ ∈ 𝔽. and do the following question from past UTA preliminary examinations: Aug 2012 #4(c). This concludes the homework assignments for the semester. Please remember to complete online the student feedback survey by 11:59 pm on Dec 2 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the test information sheets and/or the solution sheets for the tests, the examples provided in class, the JNF handout, corniness of jokes …??). Thank you!
Nov 27	THANKSGIVING HOLIDAY — see the academic schedule at https://www.uta.edu/academics/academic-calendar
Dec 02	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 11:59 pm on Dec 2 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the test information sheets and/or the solution sheets for the tests, the examples provided in class, the JNF handout, corniness of jokes …??). Thank you!
Dec 05	Online office hour today: 3:30-5:00 PM; see the Final Test’s information sheet in Canvas for the link.
Dec 07	Online office hour today: 3:00-4:30 PM; see the Final Test’s information sheet in Canvas for the link.
Dec 09	Office hour today in PKH 462: 2:30-4:00 PM.
Dec 09	FINAL TEST today, starting at 5:30 PM; check Canvas for an information sheet.
	The assignments from Fall 2022 can be viewed in their entirety here . However, note that Fall 2022’s homework was assigned using the previous edition of the textbook, so the page numbers and question numbers will differ from those of the new edition.