General information Homework Assignments
General information
Unless otherwise specified, due-dates for assignments are Tuesdays (with the exception of Assignment 0, a reading-only assignment to complete before the end of Drop/Add). Instead of homework being collected, on most due-dates there will be a quiz on the homework during the Tuesday class meeting.
The assignments and due dates are listed in the Assignments chart on this page (scroll down). For each assignment the problem-list (and other components, if any) will be updated frequently, based on how far we got in the previous lecture. Usually these updates will be made several hours after the lecture. I will NOT send out notices of these regular updates. Sometimes there will be further updates to correct typos, provide clarification, etc.
You are responsible for checking this page frequently, within a day of each lecture. This page has a "last updated" line near the top to help you tell quickly whether the assignments-chart may have changed since the last time you checked. (The "last updated" date/time will change if I update anything on this page, not just the assignments. But the vast majority of updates will be for the assignments themselves.)
Since the assignments will be built as we go along, you will see a "NOT COMPLETE YET" or "POSSIBLY NOT COMPLETE YET" notice for each assignment until that assignment's listing is complete. But you should start working on problems the day that they're added to this page (unless I say otherwise). It would be unwise to leave a week's worth of homework to the last day. The biggest mistakes students make in this course (and many others) are: not starting to work on the homework assignments early enough, and not doing all the homework. ("Doing all the homework" does not always mean succeeding with every assigned exercise; it means attempting every exercise and giving it your best effort, not giving up after a few minutes.) When I construct your exams, I'll expect you to be familiar with all the homework exercises and reading (as well as anything covered in class that might not be represented in the homework). If you fall behind, there won't be enough time later for you to catch up.
The homework quizzes will be written and graded by your TA, Jake Kowalczyk.
Some Rules for Written Work (Quizzes and Exams)
For example, never write down to the very bottom of a page. Your margins on all four sides should be wide enough for a grader to EASILY insert corrections (or comments, or partial scores) adjacent to what's being corrected (or commented on, or scored).
Academic honesty
On all work submitted for credit by students at the University of Florida, the following pledge is implied:
"On my honor, I have neither given nor received unauthorized aid in doing this assignment."
- Write in complete, unambiguous, grammatically correct, and correctly punctuated sentences and paragraphs, as you would find in your textbook.
  Reminder: Every sentence (excluding questions and exclamations) begins with a CAPITAL LETTER and ends with a PERIOD.- On every page, leave margins (left AND right AND top AND bottom; note that "and" does not mean "or").
On your quizzes and exams, to save time you'll be allowed to use the symbols \(\forall, \exists\), \(\Longrightarrow, \Longleftarrow\), and \(\iff\), but you will be required to use them correctly. The handout Mathematical grammar and correct use of terminology, assigned as reading in Assignment 0, reviews the correct usage of these symbols. You will not be allowed to use the symbols \(\wedge\) and \(\vee\), or any symbol for logical negation of a statement. There is no universally agreed-upon symbol for negation; such symbols are highly author-dependent. Symbols for and and or are used essentially as "training wheels" in courses like MHF 3202 (Sets and Logic). The vast majority of mathematicians never use \(\wedge\) or \(\vee\) symbols to mean "and" or "or"; they use \(\wedge\) and \(\vee\) with very standard different meanings. (Note: the double-arrows \( \Longrightarrow, \Longleftarrow,\) and \(\iff\) are implication arrows. Single arrows do not represent implication, so you may not use them to substitute for the double-arrow symbols.) [Depending on which Sets and Logic section you took, you may have had the misfortune to use a textbook that uses single arrows for implication. If so, you've been taught implication-notation that most of the mathematical world considers to be wrong, and, starting now, you'll need to un-learn that notation in order to avoid confusion in almost all your subsequent math courses. As an analogy: if you had a class in which you were taught that the word for "dog" is "cat", your subsequent teachers would correct that misimpression in order to spare you a lot of future confusion; they would insist that you learn that "cat" does not mean "dog". They would not say, "Well, since someone taught you that it's okay to use `cat' for 'dog', I'll let you go on thinking that that's okay."]
Date due | Assignment |
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F 1/17/25 | Assignment 0 (just reading, but important to
do before the end of Drop/Add)
I recommend also reading the handout "Taking and Using Notes in a College Math Class," even though it is aimed at students in Calculus 1-2-3 and Elementary Differential Equations. Also read the handout "Sets and Functions" on the Miscellaneous Handouts page, even though some of it repeats material in the FIS appendices. I originally wrote this for students who hadn't taken MHF3202 (at a time when MHF3202 wasn't yet a prerequisite for MAS4105), so the level may initially seem very elementary. But don't be fooled: these notes include some material that most students entering MAS4105 are, at best, unclear about, especially when it comes to writing mathematics. For the portions of my handout that basically repeat what you saw in FIS Appendices A and B, it's okay just to skim. I'd like to amplify guideline 9, "Watch out for 'it'." You should watch out for any pronoun, although "it" is the one that most commonly causes trouble. Any time you use a pronoun, make sure that it has a clear and unambiguous antecedent. (The antecedent of a pronoun is the noun that the pronoun stands for.) |
T 1/21/25 |
Assignment 1
See my Fall 2023 homework page for some notes on the reading and exercises in this assignment.
As I hope is obvious: on the Fall 2023 page, anywhere you see a statement referring to something I said in class, the thing that I "said" may be something I haven't said yet this semester, and could end up not saying this semester at all. My lectures are not word-for-word the same every time I teach this class, so you'll need to use some common sense when looking at the inserted notes in a prior semester's assignments.
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T 1/28/25 |
Assignment 2
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T 2/4/25 |
Assignment 3
Note: Two of the three procedural steps below "The procedure just illustrated" on p. 28 should have been stated more precisely. See my Fall 2023 homework page, Assignment 2, "Read Section 1.4 ..." bullet-point, for clarifications/corrections.
i.e. iff every linear combination of elements of \(W\) lies in \(W\). Once you've read the book's definition of linearly dependent and linearly independent subsets of a vector space (pp. 37 and 38), show that, together, these are equivalent to Definition 4 in my "Lists, ..." handout. Note that in the handout's Definition 4, I've emphasized the word "distinct". I've done this because when reading the book's definition on p. 37, most students don't realize how critical this word is, so they later forget it's there. The handout's Proposition 3 shows why this word is important in Definition 4, and hence also in the book's definition on p. 37. Do this exercise (which the handout's Proposition 3 should make easy): Show that if "distinct" were deleted from the handout's Definition 4, or from the book's definition on p. 37, then every nonempty subset of a vector space would be linearly dependent (and hence the terminology "linearly (in)dependent" would serve no purpose!). |
T 2/11/25 |
Assignment 4
With these changes, the results you are proving are, simultaneously, stronger than what the book asked you to prove (fewer assumptions are needed), and simpler to prove. For the (non-obvious) reason it's simpler to do the modified problem than to do the book's problem correctly, see the green text under "Modification for #13" in my Fall 2024 homework page, Assignment 4.
When you write what you think is a definition of, say, a (type of) object X or a property P, ask yourself: Would somebody else be able to tell unequivocally whether some given object is an X or has property P, using only your definition (plus any prior, precise definitions on which yours depends, but without asking you any questions)? In other words, is your definition usable? If you can't write precise definitions, you'll never be able be able to write coherent proofs. Many proofs are almost "automatic": if you write down the precise definitions of terminology in the hypotheses and conclusion, the definitions practically provide a recipe for writing out a correct proof. When you learn a new concept, a natural early stage of the learning process is to translate new vocabulary into terms that mean something to you. That's fine. But you do have to get beyond that stage, and be able to communicate clearly with people who can't read your mind. You have to do this in an agreed-upon language (English, for us) whose rules of word-order, grammar, and syntax distinguish meaningful sentences from gibberish, and distinguish from each other meaningful sentences that use the same set of words but have different meanings. If someone new to football were to ask you to tell him or her, in writing, what a touchdown is, it wouldn't be helpful to answer with, "If it's a touchdown, it means when they throw or run and the person holding that thing gets past the line." If the friend asks what a field goal is, you wouldn't answer with "A field goal is when they don't throw, but they kick, no running." But this is how limited your understanding of mathematical terminology and concepts, and your ability to use them, are likely to be if you don't practice writing definitions. For every object or property we've defined in this class, you should be able to write a definition that's nearly identical either to the one in the book or one that I gave in class (or in a handout). Until you've mastered that, hold off on trying to write the definitions in what you think are other (equivalent!) ways. Most students need these "training wheels" for quite a while. |
T 2/18/25 |
Assignment 5
   See the last item in Assignment 4 on my Fall 2023 homework page for some comments on #25.    Note: For most short-answer homework exercises (the only exceptions might be some parts of the "true/false quizzes" like 1.6/ 1), if I were putting the problem on an exam, you'd be expected to show your reasoning. So, don't consider yourself done if you merely guess the right answer! (It wouldn't hurt to review the other advice in the Some advice on how to do well and Further general advice sections of the syllabus as well.) |
T 2/25/25 |
Assignment 6
As was also true of Section 1.6, there is a lot of content in Section 2.1 (more than in any other section of Chapter 2).
In Example 8, the authors neglected to state explicitly that the two transformations in the example are linear—but they are linear, and you should show this. (That's very easy for these two transformations, but it's still a good drill in what the definition of linearity is, and how to use it.) When going through examples such as 9–11 in this section (and possibly others in later sections of the book) that start with wording like "Let \({\sf T}: {\rm (given\ vector\ space)}\to {\rm (given\ vector\ space)}\) be the linear transformation defined by ... ," or "Define a linear transformation \({\sf T}: {\rm (given\ vector\ space)}\to {\rm (given\ vector\ space)}\) by ...", the first thing you should do is to check that \({\sf T}\) is, in fact, linear. (You should do this before even proceeding to the sentence after the one in which \({\sf T}\) is defined.) Some students will be able to do these linearity-checks mentally, almost instantaneously or in a matter of seconds. Others will have to write out the criteria for linearity and explicitly do the calculations needed to check it. After doing enough linearity-checks—how many varies from person to person—students in the latter category will gradually move into the former category (or at least closer to it), developing a sense for what types of formulas lead to linear maps. In math textbooks at this level and above, it's standard to leave instructions of this sort implicit. The authors assume that you're motivated by a deep desire to understand; that you're someone who always wants to know why things are true. Therefore it's assumed that, absent instructions to the contrary, you'll never just take the author's word for something that you have the ability to check; that your mindset will NOT be anything like, "I figured that if the book said object X has property Y at the beginning of an example, we could just assume object X has property Y." | W 2/26/25 |
First midterm exam At the exam, you'll be given a booklet with a cover page that has instructions, and has the exam-problems and work-space on subsequent pages. In Canvas, under Files > exam-related, I've posted a sample cover-page. Familiarize yourself with the instructions on this page; your instructions will be similar or identical. In the same folder, the file "fall2024_exam1_probs.pdf" has the list of problems (without the workspace) that were on that exam, with some embedded comments on how the class performed.
"Fair game" material for this exam is everything
we end up covering (in class, homework, or the relevant pages of the
book) up through the Friday Feb. 21 lecture and
the homework due Feb. 25.
For this exam, and any other, the amount of material you're responsible for is far more than could be tested in an hour (or even two hours). Part of my job is to get you to study all the material, whether or not I think I'm thinking of putting this or that topic on an exam, so I generally will not answer questions like "Might we have to do such-and-such on the exam?" or "Which topics should I focus on the most when I'm studying?" My job is not to abet focusing on less material than you're supposed to be learning. If you've been responsibly doing all the assigned homework, and regularly going through your notes to fill in any gaps in what you understood in class, then studying for this exam should be a matter of reviewing, not crash-learning. (Ideally, this should be true of any exam you take in any class; it will be true of all of mine. Once again, recall my day-one advice: "Study from day one as if your next exam is next week.") Your review should have three essential components:
When reviewing work that's been graded and returned to you (e.g. a quiz), make sure you understand any comments made by the grader, even on problems for which you received full credit. There are numerous mistakes that, when made early in the semester, might get you only a warning, but that could cost you points if you're still making them later in the semester. As the semester moves along, you are expected to learn from past mistakes, and not continue making the same ones over and over. |
T 3/4/25 |
Assignment 7
Comments on some of these exercises:
1.6/ 26. Although this exercise can be done without using material from beyond Section 1.6, it can also be done as a nice application of the Rank-plus-Nullity Theorem. First show that, for any fixed \(a\in \bfr\), the map \({\rm ev}_a:P_n(\bfr) \to \bfr\) defined by \({\rm ev}_a(f)=f(a)\) is linear. (The notation I'm using comes from the name of this map: evaluation at \(a\).) Observe that the subspace whose dimension you're asked to find is exactly \(\N({\rm ev}_a)\). Then determine the range \(\R({\rm ev}_a)\) (noting that there aren't a whole lot of possibilities for a subspace of \(\bfr\) !) and apply Theorem 2.2. |
T 3/11/25 |
Assignment 8
Writing a precise definition, and understanding what you're writing, requires that you read definitions carefully. This takes TIME and CONCENTRATION. Don't multi-task, and don't just run your eyes over the words, say "Yeah, okay" to yourself, and count that as reading. Play close attention not only to what words appear, but to the order in which they appear, and to the logical and grammatical structure of the sentence(s). It is not acceptable, for example, to know only that linear combinations, spans, and linear dependence/independence have something to do with a bunch of vectors \(v_i\), a bunch of scalars \(c_i\), expressions of the form \(c_1 v_1+\dots +c_n v_n\), and sometimes the zero vector and sometimes not. Each definition of the terms above, if phrased in terms of vectors \(v_i\) and scalars \(c_i\), has to introduce and quantify those vectors and scalars; has to state (among other things) exactly what restrictions there are, if any, on the scalars and/or vectors; and has to state exactly what role the expressions "\(c_1 v_1+\dots +c_n v_n\)" play in the definition. There can be no ambiguity.
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T 3/25/25 |
Assignment 9
Powers of transformations in \(\call(\V,\V)\) and square matrices. For a linear map \(\T\) from some vector space \(\V\) to itself, the book's recursive definition of \(\T^k\) for \(k\geq 2\) is ''\(\T^k=\T^{k-1}\circ \T.\)'' An equivalent definition that I find more natural is ''\(\T^k=\T\circ\T^{k-1}\).'' (For me, the definition is, ''After you've done \(\T\) \((k-1)\) times, do it one more time,'' whereas the book's is, ''Do \(\T\), and then do it \((k-1)\) more times.'') Similarly, for a given square matrix \(A\) (recall that "square matrix" means "\(n \times n\) matrix for some \(n\)"), for me the natural recursive definition of \(A^k\) is ''\(A^k=A\, A^{k-1},\)'' rather than the book's equivalent definition ''\(A^k=A^{k-1}\, A.\)'' The book's definitions ''\(\T^0=I_\V\)'' (for any linear map \(\T:\V\to \V\)) and ''\(A^0=I_{n\times n}\)'' (for any \(n\times n\) matrix \(A\)) should be regarded as notation conventions for this book, not as standard definitions like the definition of vector space. The authors allude to this implicitly in their definition of \(\T^0\) (''For convenience, we also define ...'') but neglect to do this in their definition of \(A^0\), where the "For convenience" wording would actually be more important. After reading Theorem 2.19, go back and replace Example 5 by an exercise that says, "Show that \(P_3(\bfr)\) is isomorphic to \(M_{2\times 2}(\bfr)\)." Although that's the same conclusion reached in Example 5, there are much easier, more obvious ways to obtain that conclusion. (The point of Example 5 isn't actually to show that \(P_3(\bfr)\) is isomorphic to \(M_{2\times 2}(\bfr)\); it's to show that one way you could reach this conclusion is by using the Lagrange Interpolation Formula, something we skipped in Section 1.6 and won't be covering.) Several results in Section 2.4 are worded incorrectly. On my Fall 2023 homework page, in Assignment 8 (due-date 10/24/23), read my comments on the wording of Theorem 2.18 and its corollaries.   Notes on some of these exercises:
Please remember that all my handouts are written to help you succeed, not to burden you with extra work. (They also are/were very time-consuming to write.) So please make sure you read them, with the goal of understanding, paying enough attention while reading that you remember what you've read. (Achieving this goal may require re-reading the same thing multiple times, possibly several weeks apart. If you pay attention while reading [no multi-tasking; see https://news.stanford.edu/stories/2009/08/multitask-research-study-082409], and genuinely want to understand, the material will keep seeping into your brain without you knowing it; your brain runs programs in the background, even while you sleep. How much calendar time is needed will vary enormously from student to student.) |
T 4/1/25 |
Assignment 10
Regarding #8: we did at least half of this in class, but re-do it all to cement the ideas in your mind. Regarding 19(b): as we saw in an earlier assignment, whenever FIS asks you to "verify" a particular instance of a theorem, what the authors mean is, "Check, by direct computation, that the conclusion of the theorem is consistent with what your computation gives in this instance (or vice-versa)." Note: This textbook often states very useful results very quietly, often as un-numbered corollaries. One example of this is the corollary on p. 115, whose proof is one of your assigned exercises. There are other important results that the book doesn't even display as a corollary (or theorem, proposition, etc.), or even as a numbered equation. One example is the important matrix-product fact "\( (AB)^t=B^tA^t\) " buried on p. 89 between Example 1 and Theorem 2.11. Warning: Any version of equation (*) that you think is "mostly correct" (but isn't completely correct) is useless. Don't rely only on your memory for this formula. When you write down what you think is the inverse \(B\) of a given \(2\times 2\) matrix \(A\), always check (by doing the matrix-multiplication) either that \(AB=I\) or that \(BA=I\). (We showed in class why it's sufficient to do one of these checks.) This should take you only a few seconds, so there's never an excuse for writing down the wrong matrix for the inverse of an invertible \(2\times 2\) matrix. |
W 4/2/25 |
Second midterm exam Review the instructions on the cover page of your first exam. The instructions for the second exam will probably be identical; any changes would be minor. "Fair game" material for this exam will be everything we've covered (in class, homework, or the relevant pages of the book) up through all the material on isomorphisms, including everything in Assignment 10 other than the reading of Section 2.5. . The emphasis will be on material covered since the first midterm, but all of that relies on the earlier material, so effectively the exam is cumulative.
Failure to pick up your graded first exam or any quiz , after being absent when the graded work was returned in class, does not excuse ignorance of what mistakes of yours have been commented on or corrected. Nor does absence excuse continuing to make mistakes that I discussed in class when you were absent. You are always responsible for everything I've said in class, whether or not you were there. |
T 4/8/25 |
Assignment 11
Expect this to be a long assignment. Do not take a vacation from linear algebra after the Wednesday 4/2 midterm and postpone starting to work on this assignment. 2.5/ 1, 2, 4, 5, 6, 8, 11, 12 Comment on #6. Note that in this exercise, you are asked only to find the matrices \([L_A]_\b\) and \(Q\); you are not asked to figure out the matrix \(Q^{-1}\) or to use the formula "\([L_A]_\b=Q^{-1}AQ\)" in order to figure out \([L_A]_\b\) (which can be computed without knowing \(Q^{-1}\)—in fact, without even knowing \(Q\)). The Corollary on p. 115 tells us how to write down the matrix \(Q\) (in each part of #6) directly from the given basis \(\b\), with no computation necessary. Without even writing down \(Q\), the definition of "the matrix of a linear transformation with respect to given bases [or with respect to a single basis, for transformations from a vector space to itself]" tells everything that's needed to figure out such a matrix. For example, in 6c or 6d, letting \(w_1,w_2\), and \(w_3\) denote the indicated elements of \(\b\), we can proceed as follows:
When we want to use the formula " \([T]_{\b'}=Q^{-1}[T]_\b Q\) " (not necessary in 2.5/ 6 !) in order to explicitly compute \([T]_{\b'}\) from \([T]_\b\) and \(Q\) (assuming the latter two matrices are known), we need to know how to compute \(Q^{-1}\) from \(Q\). The above approach in green works, but is not very efficient for \(3\times 3\) and larger matrices. Efficient methods for computing matrix inverses aren't discussed until Section 3.2. For this reason, in some of the Section 2.5 exercises (e.g. 2.5/ 4, 5), the book simply gives you the relevant matrix inverse.
Note that once column rank is defined, my definition of row rank is equivalent to: \(\mbox{row-rank}(A) = \mbox{column-rank}(A^t)\). You've already covered some of Section 3.2's results in the Assignment 10's non-book problems. The version of Theorem 3.7(b) you proved in problem NB 10.3(b) is stronger than the one in the book, since the homework problem does not assume that the vector space \(Z\) is finite-dimensional. Problems NB 10.2 and 10.3(a) led you to a proof of Theorem 3.7(b) that's more fundamental and conceptual, as well as more general, than the proof in the book. Parts (c) and (d) of Theorem 3.7 then follow from parts (a) and (b), just using the book's definition of rank of a matrix (which, as I showed i Friday's class, is equivalent to my definition of column rank). Problem NB 10.3(b) (or, more weakly, Theorem 3.7ab) is an important result that has instructive, intuitive proofs that, as the homework problem shows, in no way require matrices (or anything in the book beyond Theorem 2.9). For my money, the book's proof of Theorem 3.7(b) is absurdly indirect, gives the false impression that matrix-rank needs to be defined before proving this result, further gives the false impression that Theorem 3.7 needed to be delayed until after Theorem 3.6 and one of its corollaries (Corollary 2(a), p. 156) had been proven, and obscures the intuitive reason why the result is true (namely, linear transformations never increase dimension).A note about Theorem 3.6: The result of Theorem 3.6 is pretty, and can be use to derive various other results quickly. However, the book greatly overstates the importance of Theorem 3.6; there are other routes to any important implication of this theorem. And, as the authors warn in an understatement, the proof of this theorem is "tedious to read". There's a related theorem in Section 3.4 (Theorem 3.14) that's less pretty but gives us all the important consequences that the book gets from Theorem 3.6, and whose proof is a little shorter. Rather than struggling to read the proof of Theorem 3.6, you'll get much more out of doing enough examples to convince yourself that you understand why the result is true, and why you could write out a careful proof (if you had enough time and paper). That's essentially what the book does for Theorem 3.14; the authors don't actually write out a proof the way they do for Theorem 3.6. Instead, the authors outline a method from which you could figure out a (tedious) proof. This is done in an example (not labeled as an example!) on pp. 182–184, though the example is stated in the context of solving systems of linear equations rather than just for the relevant matrix operations.
Some notes on the Section 3.2 exercises: |
T 4/15/25 |
Assignment 12
With the very poor overall performance of the class on Exam 2, I'm concerned now that by showing you previous exams of mine, I may actually be hurting your performance. That would be the case if you're (mis)using the old exams as a way of trying to guess what questions (or topics) they're likely to see on your exam, instead of studying everything you're expected to know or be able to do. I'm strongly considering not showing you a previous final exam. There are too many possible topics you're responsible for, and if I'm giving you materials that tempt you to study less, I'm making it more likely that you'll do badly on the final exam. Note: The usage of the term "the RREF of \(A\)" in Theorem 3.16 is premature, since the term does not make sense till after the corollary following the theorem is proven. For the same reason, the wording of the corollary itself is imprecise. Better wording would be "Every matrix has a unique RREF," after which we can unambiguously refer to the RREF of a given matrix. Note: Other than to understand what some assigned exercises are asking you to do, I do not care whether you know what the term "Gaussian elimination" means. I never use the term myself. As far as I'm concerned, "Gaussian elimination" means "solving a system of linear equations by (anything that amounts to) systematic row-reduction," even though that's imprecise. Any intelligent teenager who likes playing with equations could discover "Gaussian elimination" on his/her own. Naming such a procedure after Gauss, one of the greatest mathematicians of all time, is like naming finger-painting after Picasso.
Note: On p. 191, after the diamond-symbol, the authors' goal is to "streamline a procedure" that was illustrated in Section 1.6. For this reason they write the indicated homogeneous system of three equations in five unknowns; its \(3\times 6\) augmented matrix; and the RREF of this augmented matrix. But we can now accomplish the final goal—finding a subset of the five given vectors that's a basis of their span (which happens to be all of \(\bfr^3\)) with much less writing. The goal in this problem is NOT to find the solution-set of the indicated system of equations; that's the answer to a different type of question (although the questions are related). For the current problem, all we need to do is to row-reduce the 3\(\times\)5 matrix whose columns are the five given vectors, ending up with the \(3\times 5\) matrix obtained by deleting the last column of the book's matrix \(B\). Writing that never-changing all-zero 6th column, in every step from start to finish, is a waste of time (for the current problem, not if our goal were to solve the system of equations)). On the next page, in Example 3, the authors use the more efficient approach, but you have to read carefully to see that. (You should always be reading carefully, but in this particular instance, with the prominent visual display on p. 191 and the lack of anything comparable on p. 192, I think the presentation of Example 3 buries the lead. [Look up "burying the lead" if you don't know the expression.])
Determinants are important, but we simply don't have enough time to cover them the way we should. So, with regret, I won't be spending class time going over them, or proving any of their properties In Calculus 3 (and perhaps other courses) you saw how to define and compute \(2\times 2\) and \(3\times 3\) determinants, so at least you're already somewhat familiar with them. For purposes of this class, this semester, you may just take on faith that the statements in my Fall 2024 summary are true; I will expect you to know and be able to use all the properties there (except perhaps the volume-related ones). I will also expect you to know the recursive definition of \(n\times n\) determinants, and be able to compute with it. At the bottom of the Miscellaneous Handouts page, the handout "Using elementary operations to compute determinants" gives some time-saving techniques that can greatly facilitate the computation of determinants. (Note: if the way you learned to compute \(3\times 3\) determinants involved copying columns 1 and 2 to the right of column 3, then drawing certain diagonal lines, please purge that from your memory. It does not generalize to \(n\times n\) matrices with \(n>3\), and for \(n=3\) it has no time-saving advantage over the standard definition. It can't really help you, and it can harm you.) After we're done with Chapter 3, we'll cover Chapter 5 (as much as we can get to). The material there is something that you're not likely to have seen before, and it's much more important that we use whatever class time remains after Chapter 3 on Chapter 5, rather than sacrifice any Chapter-5 class time to cover Chapter 4. Chapter 5 uses determinants, however, so it's important that you know how to compute them, and what their basic properties are, by the time we start Chapter 5.
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T 4/22/25 |
Assignment 13
When you get to the Corollary on p. 247, I suggest that you read the last sentence first. That will give you more concrete idea of what a diagonalizable matrix is: A matrix \(A\) is diagonalizable iff there exists an invertible matrix \(Q\) such that \(Q^{-1}AQ\) is a diagonal matrix (equivalently: iff \(A=QDQ^{-1}\) for some invertible matrix \(Q\) and diagonal matrix \(D\)). Otherwise, in that corollary, you may get lost in the weeds, and the important last sentence may become less memorable. Once you're confident with that sentence, it's okay to go back and read the rest of the Corollary. I recommend doing 5hi by directly using the definition of eigenvector and eigenvalue rather than by computing the matrix of \({\sf T}\) with respect to a basis of \(M_{2\times 2}({\bf R})\). (I.e., take a general \(2\times 2\) matrix \(A=\left(\begin{array}{cc} a & b\\ c& d\end{array}\right) \neq \left(\begin{array}{cc} 0&0\\ 0&0\end{array}\right)\) and \(\lambda\in{\bf R}\), set \({\sf T}(A)\) equal to \(\lambda A\), and see where that leads you.) The wording of 18(d) is a good example of bad writing. The sentence should have begun with "[F]or \(n>2\)," not ended with it.
For 3f, see my recommendation above for 5.1/ 5hi. In #7, you're supposed to find an explicit formula for each of the four entries of \(A^n\), as was done for a different \(2\times 2\) matrix \(A\) in an example in Section 5.2. |
Before the final exam |
Assignment 14
| THURSDAY 5/1/25 |
Final Exam
As I mentioned in a recent email, the exam-date info on One.UF has reverted to being wrong. IGNORE ONE.UF for final-exam-date info for this class. The correct date and time, Thursday May 1 at 7:30 a.m., have always been the ones in the syllabus. |