Homework assignments and rules for written work
MAS 4105 Section 3247 (14242) — Linear Algebra 1
Fall 2024

Assignments

Date due	Assignment
W 8/28/24	Assignment 0 (just reading, but important to do before the end of Drop/Add) Note: Since Drop/Add doesn't end until after the first Tuesday class, Aug. 27, I've set the following Tuesday as the due-date for Assignment 1. But that means that Assignment 1 will include more than a week's worth of exercises and reading, all of which will be fair game for a Sept. 3 quiz. As with every homework assignment, you should start working on Assignments 0 and 1 as soon as reading or exercises start appearing for them on this page. Read the Class home page and Syllabus and course information. Read all the information above the assignment-chart on this page. Go to the Miscellaneous Handouts page and read the handouts "What is a proof?" and "Mathematical grammar and correct use of terminology". (Although this course's prerequisites are supposed to cover most of this material, most students still enter MAS 4105 without having had sufficient feedback on their work to eliminate common mistakes or bad habits.)     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. Read these tips on using your book. In FIS, read Appendix A (Sets) and Appendix B (Functions). Even though this material is supposed to have been covered in MHF3202 (except for the terminology and notation for images and preimages in the first paragraph of Appendix B that you're not expected to know yet), you'll need to have it at your fingertips. Most students entering MAS4105 don't.     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. In the book by Hammack that's the first item on the Miscellaneous Handouts page, read Section 5.3 (Mathematical Writing), which has some overlap with my "What is a proof?" and "Mathematical grammar and correct use of terminology". Hammack has a nice list of 12 important guidelines that you should already be following, having completed MHF3202. However, most students entering MAS4105 violate almost all of these guidelines. Be actively thinking when you read these guidelines, and be ready to incorporate them into your writing. Expect to be penalized for poor writing otherwise.     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 9/3/24	Assignment 1 Read Section 1.1 (which should be review). 1.1/ 1–3, 6, 7. See my Fall 2023 homework page for some notes on the reading and exercises in this assignment. As you will see from last year's homework page, prior to this semester I inserted a lot of notes into the assignments. While this had the advantage of putting those notes right in front of you, they made the assignments themselves harder to read, and some students commented in their evaluations that these notes could be overwhelming. So this year, I'm experimenting with referring you to a different page for those notes (or at least for most of them). As I hope is obvious: on last year's 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 year, and could end up not saying this year 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 last year's assignments. Read Section 1.2. Remember that whenever the book refers to a general field \(F\), you may assume \(F=\bfr\) (in this class) unless I say otherwise. If an exercise I've assigned refers specifically to the field \({\bf C}\) (plain "C " in the book) of complex numbers (e.g. 1.2/ 14) then you need to use \({\bf C}\) as the problem-setup indicates. Everything in Section 1.2 works if \(\bfr\) if replaced by \({\bf C}\); no change (other than notational) in any definition or proof is needed. 1.2/ 1, 2–4, 7, 8, 10, 11, 12–14, 17–21. Read Section 1.3. Remember: Any time I say something in class like "Exercise" or "check this" (e.g. checking that all the vector-space properties were satisfied in an example), do that work before the next class. I generally don't list such items explicitly on this homework page.
T 9/10/24	Assignment 2   Remember that you should have read Section 1.3 by now, even though (as of Wed. 9/4) I've only started covering it in class. The exercises below that involve matrices assume you've read the related portions of Section 1.3. This material about matrices is easy enough to get from reading that I'm not1 spending class time on it. We will spend class time on less easy matrix-related material later. Non-book exercise: Show that an \(n\times n\) matrix \(A\) is symmetric if and only if \(A_{ij}=A_{ji}\ \mbox{for all}\ i,j\in \{1,2\dots, n\}\). (You should find this extremely easy— you should be able to do it in your head using just the book's definition of "symmetric matrix" and the first sentence of the second paragraph on p. 18. I'm assigning mainly because it's a common way to think about what a symmetric matrix is, and because it gives a useful tool for showing that a given matrix is or isn't symmetric.) 1.3/ 1b–g, 2–10, 12–16, 18, 19, 22. [In #22, assume \(F_1= F_2=\bfr\), of course, just as I've said to assume \(F=\bfr\) when you see only a single field \(F\) in the book.]
T 9/17/24	Assignment 3   Do the following, in the order listed below. Read the first definition—the definition of \(S_1+S_2\)—near the bottom of p. 22. (The second definition is correct, but not complete; there is something else that's also called direct sum. Both types of direct sum are discussed and compared in the handout referred to below.) Exercise 1.3/ 23. In part (a), if we were to insert a period after \(V\), we'd have a sentence saying, "Prove that \(W_1+W_2\) is a subspace of \(V\)."   Think of this as part "pre-(a)" of the problem. Obviously, it's something you'll prove in the process of doing part (a), but I want the conclusion of "pre-(a)" to be something that stands out in your mind, not obscured by the remainder of part (a). Read the short handout "Direct Sums" posted on the Miscellaneous Handouts page. Do exercises DS1, DS2, and DS3 in the Direct Sums handout. Exercises 1.3/ 24, 25, 26, 28–30. See additional instructions below about 24, 28, and 30. #24: In addition to what the book says to do, figure out how #24 is related to exercise DS1. In #28, skew-symmetric is a synomym for the more commonly used term antisymmetric (which is the term I usually use). Don't worry about what a field of characteristic two is; \(\bfr\) is not such a field. See my Fall 2023 homework page for some other comments on #28. In #30, just prove the half of the "if and only if" that's not exercise DS3. In the handout Lists, linear combinations, and linear independence (also posted on the Miscellaneous Handouts page), read Definitions 1, 2, 4, and 5. (The handout was originally written for a class that had already covered Sections 1.4, 1.5, 1.6, and 2.1 of FIS. Items in the handout that aren't included in the current reading assignment will be added at an appropriate time.) Also read the "Some additional comments" section at the end of the handout. Our textbook defines linear combination and span (and, later, linear dependence/independence) primarily for sets of vectors—subsets \(S\) of a vector space \(V\)—and addresses lists of vectors only in a rather understated way (and without giving a name to them). The definitions for lists are given implicitly, and with some ambiguities, at various points in the book, e.g. in the second sentence of "Definitions" at the top of p. 25. The book's definitions of everything related to linear combinations are all correct, and are equivalent to the ones in my handout. However, for a subset \(S\subseteq V\) that has infinitely many vectors, the fact that the elements of \(S\) can't all be listed leads to some subleties that are easily overlooked (even by the authors, in at least one exercise) when using the book's definitions. This, in turn, makes it easier for students to make mistakes in the statements and/or proofs of various facts. In addition, the way we conceptualize the notions of linear combinations and linear dependence/independence is virtually always in terms of lists. It helps to have definitions that are based more directly on the way we think about linear combinations and related concepts. Read Section 1.4, minus Example 1.       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 for clarifications/corrections. 1.4/ 3abc, 4abc, 5cdegh, 10, 12, 13, 14, 17. In 5cd, the vector space in consideration is \(\bfr^3\); in 5e it's \(P_3(\bfr)\); in 5gh it's \(M_{2\times 2}(\bfr)\). Note on #12. For nonempty sets \(W,\) the way I think of the fact proven in #12 is: In a vector space \(V\),   a nonempty subset   \(W\subseteq V\)   is a subspace iff   \(W\)   is "closed under taking linear combinations;" i.e. iff every linear combination of elements of   \(W\)   lies in   \(W\). The book's wording in #12 is better in a couple of ways: (i) it handles the empty-set case as well as the nonempty-set case, and (ii) it is very efficient. However, our minds don't always conceptualize things in the most efficient way.   My less efficient phrasing indicates more directly what I'm thinking (in this context), and avoids one extra word of recently learned vocabulary: span.   (But this comes at a cost: introducing other new terminology—"closed under taking linear combinations", a non-standard term that itself requires definition—as well as not handling the empty-set case.)
T 9/24/24	Assignment 4   In the handout Lists, linear combinations, and linear independence, do Exercise 6 and read Example 7. Read Section 1.5 1.5/ 1, 2(a)–2(f), 3–7, 10, 12 (done in class; re-do for practice), 13 (modified as below), 15, 16, 17, 20. (Reminder: as mentioned in the previous assignment, ignore any instructions related to the characteristic of a field. Just FYI, without giving you a definition, the field \(\bfr\) happens to have characteristic 0.) Modification for #13. Erase all the set-braces and insert the words "the list" in front of "\(u, v\)",   "\(u+v, \ u-v\)",   "\(u, v, w\)", and "\(u+v,\ u+w,\ v+w\)". Furthermore, in part (a), do not assume that the vectors \(u\) and \(v\) are distinct; in part (b), do not assume that \(u, \ v\), and \(w\) are distinct.     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. The reason it's simpler to do the modified problem than to do the book's problem correctly is that the book's notation in #13 slides something under the rug: the book puts each list of vectors inside set-braces, giving the impression that the terms of the list are distinct, and thus the impression that the \(u+v\) and \(u-v\)   are distinct in part (a), and that all three vectors \(u+v,\)   \(u+w\),   and \(v+w\)"   are distinct in part (b). But, in each case, we are not given that these listed vectors are distinct, so we can't assume they're distinct. Thus, to do the book's problem correctly, your argument would have to include extra steps to establish distinctness: In part (a), for the direction of the "iff" in which you assume that the set \(\{u, v\}\) is linearly independent, you would have to check that the vectors \(u+v\) and \(u-v\) are distinct before trying to draw any conclusions from an equation like "\(a_1(u+v)+a_2(u-v)=0_V.\)" Checking distinctness is not hard to do, but it's a step. No such step is needed in the modified version because, from the handout's Proposition 3b, linear independence of a list of vectors implies distinctness of the terms. In part (b), for the direction of the "iff" in which you assume that the set \(\{u,v,w\}\) is linearly independent,you would have to check that all three vectors \(u+v,\ u+w,\ v+w\) are all distinct before trying to draw any conclusions from an equation like "\(a_1(u+v)+a_2(u+w)+a_3(v+w)=0_V .\)" It's again not hard to check that they are distinct; it's just an annoying extra step that's unnecessary in the "list" version of this problem. Also note that for this particular list \(u+v, u+w, v+w\), the arguments that (i) \(u+v\neq u+w\), (ii) \(u+v\neq v+w\), and (iii) \(u+w\neq v+w,\)   are all essentially the same, which reduces the amount of work needed for the extra "check distinctness" step. This extra step would require more work if instead of fine-tuned list "\(u+v,\ u+w,\ v+w,\)"   the book had given you anything even a little different, e.g. the list "\(u+v+w,\ v+w,\ w\)" (for which the same "iff" statement is still true). For the same direction of the "iff" as above, but with this new list of vectors, one possibility you'd have to rule out is "\(u+v+w=w\)," which you can't do using exactly the same argument by which it's easiest to rule out "\(u+v+w=v+w\)" and "\(v+w=w\)". In Section 1.6, read from the beginning up through Theorem 1.8 and its (partial) proof. (Only one direction of the "if and only if" is proved—specifically, the "only if" direction [Make sure you understand that for a birectional implication phrase as "P if and only if Q", the "if" direction is "P if Q", i.e. "Q implies P." The implication "P implies Q" is the "only if" direction.]) The argument in the book should look very familiar, because we used the exact same argument to prove an equivalent result in class on Friday 9/21; we just hadn't introduced the word "basis" yet. What we proved in class was: if \(v_1, \dots, v_n\) is a linearly independent list \(L\) in a vector space \(V\), then each \(v\in \span(L)\) can be expressed as \(c_1v_1+\dots+c_n v_n\) for a unique list of scalars \(c_1, \dots, c_n\). But the terms of a linearly independent list are automatically distinct, so, as noted in class and in my handout, a linearly independent list \(v_1, \dots, v_n\) can be identified with the ordered \(n\)-element linearly independent set \(S=\{v_1, \dots, v_n\}\) just by inserting the set-braces. Clearly \(\span(S)=\span(L)\). Thus if \(\span(S)=V\)—i.e. if \(S\) is a basis of \(V\)  —  then the proposition proven in class yields what's proven in the book. It might appear at first that the "unique representability" result we proved in class is more general than the "only if" direction of Theorem 1.8, since in class we didn't demand that \(\span(S)=V\). But that "extra generality" is illusory. By definition, every subset of a vector space \(V\) spans its own span ("\(S\) spans \(\span(S)"\)). Thus every linearly independent set in \(V\) is a basis of its own span—which is a subspace of \(V\), hence a vector space. So the "only if" half of Theorem 1.8 implies the (superficially more-general-looking) "unique representability" result we proved in class for vectors in the span of a linearly independent list.
T 10/1/24	Assignment 5 (identical to what was incorrectly inserted as part of Assignment 4 on 9/27) In whichever order you prefer, finish reading Section 1.6—minus the subsection on the Lagrange Interpolation Formula—and read the handout Some Notes on Bases and Dimension. Although you may choose which to read first, finish either the Section 1.6 reading or the handout before the Monday 9/29 class. These two readings have a lot of overlap (covered in different orders), but neither can substitute entirely for the other. (However, whichever you read second, it's okay just to skim anything you thoroughly understood from your first reading.) There is some material in my handout that's not covered in Section 1.6, but my handout has very few of the examples that are in Section 1.6. Some proofs and other items in FIS that might give you difficulty have expanded versions in the handout.    Because of the lecture we lost to Hurricane Helene, I will not have time to cover in class everything that's in the handout, but, as with any assigned reading, everything there is fair game for an exam. 1.6/ 1–8, 12, 13, 17, 21, 25 (see below), 29, 33, 34. The last page of my handout has a summary that includes various facts that can (and should) be used to considerably shorten the amount of work needed for several of these exercises, e.g. #4 and #12.     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!
T 10/8/24	Assignment 6 1.6/ 14–16, 18 (note: in #18, \({\sf W}\) is not finite-dimensional!), 22, 23, 30, 31, 32 Do these non-book problems. Read Section 2.1 up to, but not including, the "Definitions" paragraph near the bottom of p. 67. (Originally I said to read up through Theorem 2.2. I have adjusted the assignment to reflect how far we got in Friday's class, since that's the cutoff for Wednesday's exam.) Rephrase what's being proven in exercise 1.3/ 3 (from Assignment 2) as a statement that a certain map from some (specific) vector space to another is linear. 2.1/ 2–6 (only the "prove that \({\sf T}\) is a linear transformation" part, for now), 7–9, 12.     Regarding #7: we proved properties 1 and 2, so those parts of the exercise will be review. Note that there is actual work for you to do when reading many of the examples in Section 2.1. In Section 2.1, Example 1 is essentially the only example in which the authors go through all the details of showing that the function under consideration is linear. In the remaining examples, the authors assume that all students can, and therefore will, check the asserted linearity on their own. Examples 2–4 are preceded by a paragraph asserting that the transformations in these examples are linear, and saying, "We leave the proofs of linearity to the reader"—meaning you!     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."
M 10/14/24	First midterm exam (postponed from 10/2, partly thanks to Hurricanes Helene and Milton) 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 files, I've posted a sample cover-page ("exam cover-page sample.pdf"). Familiarize yourself with the instructions on this page; your instructions will be similar or identical. In the same folder, the file "fall2023_exam1_probs.pdf" has the list of problems that were on that exam (no workspace, just the list).       "Fair game" material for this exam is everything we've covered (in class, homework, or the relevant pages of the book) up through the Friday Oct. 4 lecture and the homework due Oct. 8.     In FIS Chapter 1, we did not cover Section 1.7 or the Lagrange Interpolation Formula subsection of Section 1.6. (However, homework includes any reading I assigned, which includes all handouts I've assigned. (For example, fair-game material includes everything in my handout on bases and dimension, even though the "maximal linearly independent set" material overlaps with the book's Section 1.7.) You should regard everything else in Chapter 1 as having been covered (except that the only field of scalars we've used, and that I'm holding you responsible for at this time, is \(\bfr\)).       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 it's going to end up 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?"       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; it will be true of all of mine.) Your review should have three components: reviewing your class notes; reviewing the relevant material in the textbook and in any handouts I've given; and review the homework (including any reading not mentioned above). If you're given an old exam to look at, then of course you should look at that too, but that's the tip of the iceberg; it does not replace any of the review-components above (each of which is more important), and cannot tell you how prepared you are for your own exam. Again, on any exam, there's never enough time to test you on everything you're responsible for; you get tested on a subset of that material, and you should never assume that your exam's subset will be largely the same as the old exam's subset.       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 for which you may get away with only a warning earlier in the semester, 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 to make them.
T 10/15/24	Assignment 7 Read Section 2.1 from where you left off up through Example 13. 2.1/ 1a–1f, 2–6 (the parts not done for previous assignment) 10, 11, 13–18, 20 For 1f, look at #14a first. In 2–6, one thing you're asked to determine is whether the given linear transformation \( {\sf T:V}\to {\sf W}\) is onto. In all of these, \(\dim(V)\leq \dim(W)\). This makes these questions easier to answer, for the following reasons: If \(\dim({\sf V})<\dim({\sf W})\), then \({\sf T}\) cannot be onto; see exercise 17a. When \(\dim(V)=\dim(W)\), we may be able to show directly whether \({\sf T}\) is onto, but if not, we can make use of Theorem 2.5 (when \(\dim(V)=\dim(W)\), a linear map \({\sf T}: {\sf V}\to {\sf W}\) is onto iff \({\sf T}\) is one-to-one). We can determine whether \({\sf T}\) is one-to-one using Theorem 2.4. Also, regarding the "verify the Dimension Theorem" part of the instructions: You're not verifying the truth of the Dimension Theorem; it's a theorem. What you're being asked to do is to check that your answers for the nullity and rank satisfy the equation in Theorem 2.3. In other words, you're doing a consistency check on those answers. (This comment, added Oct. 14, is the only change to Assignment 7 since Oct. 10.) In #10: For the "Is \({\sf T}\) one-to-one?" part, you'll want to use Theorem 2.4, but there's more than one way of setting up to use it. You should be able to do this problem in your head (i.e. without need for pencil and paper) by using Theorem 2.2, then Theorem 2.3, then Theorem 2.4. In 14a, the meaning of "\({\sf T}\) carries linearly independent subsets of \( {\sf V} \) onto linearly independent subsets of \( {\sf W} \)" is: if \(A\subseteq {\sf V}\) is linearly independent, then so is \({\sf T}(A)\). For the notation "\({\sf T}(A)\)", see the note about #20 below. In #20, regarding the meaning of \({\sf T(V_1)}\): Given any function \(f:X\to Y\) and subset \(A\subseteq X\), the notation "\(f(A)\)" means the set \( \{f(x): x\in A\} \). (I discussed this in class on Monday 10/7/24. If you've done all your homework, you already saw this in Assignment 0; it's in the first paragraph of FIS Appendix B.) The set \(f(A)\) is called the image of \(A\) under \(f\). For a linear transformation \({\sf T}:{\sf V}\to {\sf W}\), this notation gives us a second notation for the range: \({\sf R(T)}={\sf T(V)}\).
T 10/22/23	Assignment 8 Practice writing definitions!!! If you can't write precise definitions, you'll never be able be able to write coherent proofs.     For each 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 of you will need these "training wheels" for a while.     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."     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, given some object, be able to tell unequivocally whether that object is an X or has property P, using only your definition alone (plus any prior, precise definitions on which yours depends, but without asking you any questions)? In other words, is your definition usable?     Writing a precise definition requires that you read definitions carefully. Play close attention not just 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 \(c_i\), and expressions of the form \(c_1 v_1+\dots +c_n v_n\). 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. You can't build without firm building-blocks. From now on, an implicit part of every homework assignment is: For every definition given in class or in assigned reading, practice writing the definition without looking at the book or any notes until you're able to reproduce the definition you were given. Read the remainder of Section 2.1. Read Section 2.2. 2.1/ 1gh, 21, 22 (just the first part), 23, 25, 27, 28, 36. See comments below on some of these exercises. #25: In the definition at the bottom of p. 76, the terminology I use most often for the function \({\sf T}\) is the projection [or projection map] from \({\sf V}\) onto \({\sf W}_1\). There's nothing wrong with using "on" instead of "onto", but this map \({\sf T}\) is onto. I'm not in the habit of including the "along \({\sf W}_2\)" when I refer to this projection map, but there is actually good reason to do it: it reminds you that the projection map depends on both \({\sf V}\) and \({\sf W}\), which is what exercise 25 is illustrating. #28(b): If you've done the assigned exercises in order, then you've already seen such an example. #36: Recall that the definition of "\({\sf V}\) is the (internal) direct sum of two subspaces \({\sf V_1, V_2}\)" had two conditions that the pair of subspaces had to satisfy. Problem 36 says that, when \({\sf V}\) is finite-dimensional and the subspaces are the range and the null space of the same linear map, each of these conditions implies the other. Consequently, for a linear map \({\sf T}: V\to V\), where \(V\) is a finite-dimensional vector space, you only have to verify one of these conditions in order to conclude that \({\sf V=R(T)\oplus N(T)}\). This is reminiscent of some other instances we've seen of "things with two conditions" for which, under some hypothesis, each of the conditions implied the other. For example: A set of \(S\) of \(n\) vectors in an \(n\)-dimensional vector space \({\sf V}\) is linearly independent if and only if \(S\) spans \(V\). (Hence \(S\) is a basis of \({\sf V}\) if either condition is satisfied.) Given two vector spaces \({\sf V}, {\sf W}\) of equal (finite) dimension, a linear map \({\sf T: V\to W}\) is one-to-one if and only if \({\sf T}\) is onto. Do these non-book problems.
T 10/29/24	Assignment 9 2.2/ 1–7, 12, 16a (modified as below), 17 (modified as below). In #16a: Show also (not instead of) that an equivalent definition of \({\sf S}^0\) is: \({\sf S^0= \{ T\in {\mathcal L}(V,W): N(T)\supseteq {\rm span}(S)\}} \). In #17: Assume that \({\sf V}\) and \({\sf W}\) have finite, positive dimension (see note below). Also, extend the second sentence so that it ends with "... such that \([{\sf T}]_\beta^\gamma\) is a diagonal matrix, each of whose diagonal entries is either 1 or 0." (This should actually make the problem easier!)     Additionally, show that if \({\sf T}\) is one-to-one, and the bases \(\beta,\gamma\) are chosen as above, none of the diagonal entries of \([{\sf T}]_\beta^\gamma\) is 0. (Hence they are all 1, and \([{\sf T}]_\beta^\gamma\) is the \(n\times n\) identity matrix \(I_n\) defined on p. 82, where \(n=\dim(V)=\dim(W)\).) Note: Using a phrase like "for positive [something]" does not imply that that thing has the potential to be negative! For example, "positive dimension" means "nonzero dimension"; there's no such thing as "negative dimension". For numerical quantities \(Q\) that can only be positive or zero, when we don't want to talk about the case \(Q=0\) we frequently say "for positive \(Q\)", rather than something like "for nonzero \(Q\)". Read Section 2.3. 2.3/ 2, 4–6, 11–13.     In #11, remember that \({\sf T}_0\) is the book's notation for the zero linear transformation (also called "zero map") from any vector space \(V\) to any any vector space \(W\). As I've mentioned in class, I'd have preferred notation such as \({\sf 0_V^W}\), or at least \(0_{\rm fcn}\) or \(0_{\rm map}\), for the zero map from \(V\) to \(W\). In Canvas, under Files (folder "exam-related files"), find and read these handouts: fall2024_exam1_solutions.pdf fall2023_exam1_comments1.pdf fall2023_exam1_comments2.pdf Although I posted the two 2023 files more than a week before your exam, and announced the posting in an Oct. 6 email, most or all of mistakes discussed there were still made by several students on your exam. Review some previously assigned readings, as follows: In the Mathematical grammar and correct use of terminology handout (whose reading was part of Assignment 0), review the "Some common mistakes" section. Many of these mistakes were made by some student(s) in this class on the first exam. (The listed mistakes that weren't made on the first exam have to do, primarily, with material we haven't covered yet.) If I corrected any sentence-structure on your exam (including capitalizing a letter at the beginning of a sentence, and ending a sentence with a period), or commented on your sentences—or lack thereof—or your English or writing (including one-word comments like "English!" or "Writing!"), review these parts of the What is a proof? handout: (i) from the beginning, up through the first paragraph on p. 2, and (ii) the "Some pitfalls ..." section on pp. 3–6. 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 11/5/24	Assignment 10   2.3/ 1, 7, 8, 14, 15, 16a, 17, 18. Notes on several of these exercises: In 1e, it's implicitly assumed that \(W=V\); otherwise the transformation \({\sf T}^2\) isn't defined. Similarly, in 1f and 1h, \(A\) is implicitly assumed to be a square matrix; otherwise \(A^2\) isn't defined. In 1(i), the matrices \(A\) and \(B\) are implicitly assumed to be of the same size (the same "\(m\times n\)"); otherwise \(A+B\) isn't defined. In 2a, make sure you compute  \( (AB)D\)   *AND*   \(A(BD)\)   as the parentheses indicate. DO NOT ASSUME, OR USE, ASSOCIATIVITY OF MATRIX-MULTIPLICATION IN THIS EXERCISE. The whole purpose of exercise 2 is for you to practice doing matrix-multiplication, not to practice using properties of matrix-multiplication. If your computations are all correct, you'll wind up with the same answer for \(A(BD)\) as for \((AB)D\). #18: See comments on my Fall 2023 homework page, Assignment 7 (due-date 10/17/23). (Not important.) In #14, you might wonder, "Why are they defining \(z\) to be \((a_1, a_2, \dots, a_p)^t\) instead of just writing   \(z=\left( \begin{array}{c}a_1\\ a_2\\ \vdots \\ a_p\end{array}\right) \)  ? "   Historically, publishers required authors to write a column vector as the transpose of a row vector, both because it was harder to typeset a column vector than a row vector and because the column vector used more vertical space, hence required more paper. I can't be sure whether those were reasons for the book's choice in this instance, but it's possible. Other possible reasons are (i) it's a little jarring to see a tall column vector in the middle of an otherwise-horizontal line of text, and (ii) the fact that in LaTeX (the mathematical word-processing software used to typeset this book) it takes more effort to format a column vector than a row vector. Read Section 2.4, skipping Example 5.     After reading Theorem 2.19, go back and replace Example 5 by an exercise that says, "\(P_3(\bfr)\) 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 get there. Read the comments about this on my Fall 2023 homework page, Assignment 8 (due-date 10/24/23).     In the same Fall 2023 assignment, also read my comments on the wording of Theorem 2.18 and its corollaries. 2.4/ 2, 3, 13, 15, 17 (with 17b modified; see below), 23. Some notes on these exercises: In #2, keep Theorem 2.19 in mind to save yourself a lot of work. Regarding 17a: in a previous homework exercise (2.1/ 20), you already showed that the conclusion holds if \(\T\) is any linear transformation from \(\V\) to \(\W\); we don't need \(\T\) to be an isomorphism. Modify 17b by weakening the assumption on \(\T\) to: "\(\T:\V\to\W\) is a one-to-one linear transformation." (So, again, we don't need \(\T\) to be an isomorphism, but this time we do need more than just "\(\T\) is linear.") Regarding #23: The book's notation makes even me say "Huh?" To unravel the notation, you have to go back not just to exercise 1.6/ 18, but to Section 1.5, Example 2—at which point you may notice that what's being called a sequence in exercise 2.4/ 23 is not consistent with the definition of "sequence" in the Section 1.5 example. (The sequences in 2.4/ 23 are functions from the set of non-negative integers to \(\bfr\), rather than from the set of positive integers to \(\bfr\).)     There are a couple of ways to fix this, and to write the definition of \(T\) more digestibly. One such way is this: Leave the definition of "sequence" in Section 1.5 Example 2 unchanged (with 1 being the initial index of every sequence), but instead of notation such as \(\sigma\) for a sequence, use notation such as \(\vec{a}\) for the sequence whose \(n^{\rm th}\) term is \(a_n\) (for every \(n\geq 1)\). Then, in exercise 2.4/ 23, define \(\T\) by "\(\T(\vec{a}) = \sum_{n=0}^{N-1} a_{n+1} x^n\), where \(N\) is the largest integer such that \(a_N\neq 0\)." 2.3/ "13\(\frac{1}{2}\)". (This extension of 2.3/ 13 really should have been part of the previous assignment.) (a) Returning to exercise 2.3/ 13 from the last assignment, now let \(A\) and \(B\) be matrices of sizes \(m\times n\) and \(n\times m\) respectively, where \(m\) and \(n\) may or may not be equal. If \(m\neq n\), then \({\rm tr}(A)\) is not defined, but both \(AB\) and \(BA\) are square matrices (of sizes \(m\times m\) and \(n\times n\) respectively), so their traces are defined. Show that \({\rm tr}(AB)={\rm tr}(BA)\) whether or not \(m=n\). For a consistency check on this remarkable property, choose a \(2\times 3\) matrix \(A\) and a \(3\times 2\) matrix \(B\), compute \(AB\) and \(BA\), and check that the traces are indeed equal.     (b) Check that if matrices \(A, B, C\) are of sizes for which the products \(ABC\) and \(BCA\) are defined, then the product \(CAB\) is also defined. Then use part (a) to show that, in this instance, \({\rm tr}(ABC)={\rm tr}(BCA)={\rm tr}(CAB)\). This is often called the "cyclic property of the trace".     (c) Show that the cyclic property of the trace generalizes to products of any number of compatibly sized matrices.
T 11/12/24	Assignment 11 Do these non-book problems. 2.4/ 1, 4–9, 14, 16, 19. Regarding #8: we did most of this in class, but re-do it all to cement the ideas in your mind. Read Section 2.5.     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 matrix-product fact   "\( (AB)^t=B^tA^t\)"   buried on p. 89 between Example 1 and Theorem 2.11. 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 \(Q^{-1}\)). 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. 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: Compute \(L_A(w_1)\) (which is simply \( Aw_1\)). Express \(Aw_1\) as a linear combination of \(\{w_1,w_2,w_3\}\)—thus, as \(c_1w_1+c_2w_2+c_3w_3\)  for some \((c_1,c_2,c_3)\)—by solving the appropriate system of three equations in three unknowns, as you were doing in various exercises in Chapter 1. These coefficients \((c_1,c_2,c_3)\) form the first column of \([L_A]_\b\). Now repeat with \(Aw_2\) and \(Aw_3\) to get the second and third columns of \([L_A]_\b\). If we did want to compute \(Q^{-1}\)—the matrix that expresses the standard basis vectors \(e_1, e_2,\) and \(e_3\) in terms of \(\beta\)—we could do that by going through steps 2, 3, and 4 of the procedure above, but with \(Aw_i\)  replaced by \(e_i\).     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 blue 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.     But inverses of \(2\times 2\) matrices arise so often that you should eventually find that you know the following by heart (like the way you know your Social Security number withot ever trying to memorize it): the matrix \(A=\abcd\) is invertible if and only if \(ad-bc\neq 0\), in which case $$ \abcd^{-1}= \frac{1}{ad-bc} \left( \begin{array}{rr} d&-b\\ -c&a \end{array}\right).\ \ \ \ (*) $$ (You should check, now, that if \(ad-bc\neq 0\), and \(B\) is the right-hand side of (*), then \(AB=I=BA\) [where \(I=I_{2\times 2}\)], verifying the "if" half of the "if and only if" and the formula for \(A^{-1}\). All that remains to show is the "only if" half of the "if and only if". You should be able to work out a proof of the "only if" on your own already, but I'm leaving that for a later lecture or exercise.) Warning: Any version of (*) 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. You're showing this again in problem NB 11.7.) 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 11/13/24	Second midterm exam Review the general comments (those not related to specific content) posted on this page for the first 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 the Friday Nov. 8 class and the complete Assignment 11. At the time I'm posting this (Thursday night, Nov. 7), I'm estimating tha the cutoff will be either the middle or end of Section 2.5. The emphasis will be on material covered since the first midterm. Reminder: As the semester moves along, your mathematical writing is expected to improve. You are expected to have learned from corrections made on your graded quizzes and exams, or were addressed in class. Various mistakes that may not have cost many (or any) points earlier in the semester will be more costly now.
T 11/19/24	Assignment 12 Read Section 3.1. Do non-book problem NB 11.3(b), which was accidentally omitted from the original NB 11 list, but logically belongs there rather than in a new group of non-book problems. 3.1/ 1, 3–8, 10, 11. Some notes on these problems: 1(c) is almost a "trick question". If you get it wrong and wonder why, the relevant operation is of type 3. Note that in the definition of a type 3 operation, there was no requirement that the scalar be nonzero; that requirement was only for type 2. In #7 (proving Theorem 3.1), you can save yourself almost half the work by (i) first proving the assertion just for elementary row operations, and then (ii) applying #6 and #5 (along with the fact "\((AB)^t=B^tA^t\) " stated and proven quietly on p. 89). In #8, I don't recommend using the book's hint, which essentially has you repeating labor done in #7 instead benefiting from the fruits of that labor. Instead I would just use the result of #7 (Theorem 3.1) and Theorem 3.2. (Observe that if \(B\) is an \(n\times n\) invertible matrix, and \(C,D\) are \(n\times p\) matrices for which \(BC=D\), we have \(B^{-1}D= B^{-1}(BC)=(B^{-1}B)C=IC=C,\) where \(I=I_{n\times n}\). [Note how similar this is to the argument that if \(c,x,y\) are real numbers, with \(c\neq 0\), the relation \(y=cx\) implies \(x=\frac{1}{c}y = c^{-1}y\). Multiplying a matrix on the left or right by an invertible matrix (of the appropriate size) is analogous to dividing by a nonzero real number. But in the matrix case, we don't call this operation "division".]) Read Section 3.2, except for (i) the statement and proof of Corollary 1 (which isn't important enough to be the best use of your time) and (ii) the proof of Theorem 3.7. If you haven't yet done non-book problem NB 11.3(b), do it now; it establishes parts (a) and (b) of Theorem 3.7.     The version of Theorem 3.7(b) in the homework problem is stronger than the one in the book, since the homework problem does not assume that the vector space \(Z\) is finite-dimensional. The proof-strategy for Theorem 3.7ab that I'm trying to lead you to with problems NB 11.2 and NB 11.3(a) is 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 definition of rank of a matrix.     Problem NB 11.3(b) (or, more weakly, Theorem 3.7ab) is an important result that has instructive, intuitive proofs that 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) were 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 it's true that it 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 consequence 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. 3.2/ 1–3, 5 (the "if it exists" should have been in parentheses; it applies only to "the inverse", not to "the rank"), 6(a)–(e), 11, 14, 15, 21, 22. Some notes on these exercises: In #6, one way to do each part is to introduce bases \(\beta, \gamma\) for the domain and codomain, and compute the matrix \([T]_\beta^\gamma\). Remember that the linear map \(T\) is invertible if and only if the matrix \([T]_\beta^\gamma\) is invertible. (This holds no matter what bases are chosen, but in this problem, there's no reason to bother with any bases other than the standard ones for \(P_2({\bf R})\) and \({\bf R}^3\).) One part of #6 can actually be done another way very quickly, if you happen to notice a particular feature of this problem-part, but this feature might not jump out at you until you start to a compute the relevant matrix. Exercises 21 and 22 can be done very quickly using results from Assignment 11's non-book problems. (You figure out which of those problems is/are the one(s) to use!) In Section 3.3, read up through Example 6.
Tues. 11/26/24 for the reading; Thurs. 11/28/24 for the exercises (target dates to help you pace yourself)	Assignment 13a   -------------------------------------------------------------- Before the Friday 11/22 lecture: If you were in class on Wed. 11/20, skim from the start of Section 3.4 through the sentence on p. 187 that begins, "Notice that we have ignored ..."; everything important in this part of Section 3.4 was covered in class on Wed. (If you were not in class on Wed., then read these pages in detail. Whether or not you were in class on Wed, 11/20, read from the next sentence on p. 187 through the end of the proof of Theorem 3.15 on p. 189. Although I went over some of this in class, there were a few importance points I didn't get to. -------------------------------------------------------------- Read the remainder of Section 3.4 (the subsection "An Interpretation of the Reduced Row Echelon Form"). Much of this was covered in Wednesday's class implicitly, but there are several things I didn't state explicitly.     For our purposes in this class, the corollary following Theorem 3.16 is not important, other than giving us the convenience of saying "the RREF" of a matrix \(A\) rather than "a  RREF of \(A\)."     Note that 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. 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. Read Chapter 4, omitting Section 4.5. In Section 4.3, it's okay if just skim the portion from Theorem 4.9 through the end of the section. I am not holding you responsible for the formula in Theorem 4.9. (Cramer's Rule is just this formula, not the whole theorem. You are responsible for knowing, and being able to show, that if \(A\) is invertible, then \(A{\bf x}={\bf b}\) has a unique solution, namely \(A^{-1}{\bf b}.\)) 3.4/ 1, 2, 7–13. Theorem 3.16(c) is key to the 7–13 group; re-read from the diamond-symbol on p. 191 through p. 193 if you're having trouble with these. 3.3/ 1–5, 7–10. In #9, there is practically nothing to do; the point of the exercise is to help you realize that the definition of "\(Ax=b\) has a solution" is exactly the the same as the definition of "\(b\in \sfr(\sfl_A)\)" combined with the definition of \(\sfl_A\).
T 12/3/24	Assignment 13b 4.1/ 1, 2, 4–9. Do not use results from later sections of Chapter 4 to help with any of these. The point of doing 5–9 is to see why the formula for \(2\times 2\) determinants (and computations using this formula, in the case of #9)  directly  implies the results of these exercises. This makes it easier, later, to remember and understand the corresponding results for general \(n\times n\) determinants later in the chapter. 4.2/ 1–3, 5, 8, 11, 23–25, 27, 29. As you may notice when doing 4.2/ 1, in Chapter 4 (and occasionally in other chapters), some parts of exercise 4.(n+1)/ 1 duplicate parts of exercise 4.n/ 1. Do as you please with the duplicates: either skip them, or use them for extra practice. 4.3/ 1a–1f, 9–12, 15, 19. (For the odd-\(n\) case of #11, you should find that 4.2/25 is a big help.) In #15, for the definition of similar matrices, see p. 116.) Read my own summary of some facts about determinants below. (The title of Section 4.4 is somewhat misleading. The book's "summary" omits many important facts, and intersperses its summarized facts with uses of these facts (so that the summarized facts don't appear in a single list.) The (unlabeled) examples on pp. 233–235 are useful, instructive, and definitely worth reading, but hardly belong in a summary of  facts   about determinants. 4.4/ 1, 4ag.     If I were asked to do 4g, I would probably not choose to expand along the second row or fourth column. Do you see why? If you were asked to compute   \(\left| \begin{array}{cc} 1 & 2 & 3\\ 0& 0 & 4 \\ 5&6&7\end{array}\right|, \)   which method would you use? ----------------------------------------------------------------------- Summary of some facts about determinants In this summary, every matrix \(A, B, \dots,\) is \( n\times n\), where \(n\geq 1\) is fixed but arbitrary (unless otherwise specified). The following are equivalent: \({\rm rank}(A)=n.\) The set of columns of \(A\) is linearly independent. The set of columns of \(A\) is a basis of \({\bf R}^n\). The set of rows of \(A\) is linearly independent. The set of rows of \(A\) is a basis of \({\bf R}^n\). \(A\) is invertible. \(\det(A)\neq 0.\) \(\det(I)=1\)  (where \(I\) is the \(n\times n\) identity matrix) \(\det(AB)=\det(A)\, \det(B)\) If \(A\) is invertible, then \(\det(A^{-1})=1/\det(A). \) If \(A\) and \(B\) are similar matrices, then \(\det(A)=\det(B)\)   (assigned exercise 4.3/ 15). \(\det(A)=\det(A^t)\) Determinant is a multilinear function of the columns (respectively, rows) of an \(n\times n\) matrix, meaning: if any \(n-1\) of the columns (resp., rows) of the \(n\times n\) matrix are held fixed, and we allow the other column (rep., row) to vary over \(\bfr^n\), and take the determinant of the matrix with this one variable column (resp., row) and the others held fixed, the resulting function from \(\bfr^n\) to \(\bfr\) is linear. We sometimes express this by saying the determinant function is "linear in each column [or row] separately." (For example, if \( w_2, \dots, w_n\) are fixed column vectors in \(\bfr^n\), then the function \(\T:\bfr^n\to\bfr\) defined by \(\T(v)= \det\left( \begin{array} {c|c|c|c} v & w_2 & \dots & w_n \end{array} \right) \) is a linear transformation. The analogous statement holds whichever column we allow to vary, holding the others fixed, and the analogous statements with "columns" replaced by "rows" are true as well. For a more detailed discussion of multilinearity, see these non-book problems on multilinearity from an earlier semester.) In particular, if \(A'\) is a matrix obtained from \(A\) by multiplying exactly one column or row of \(A\) by a nonzero real number \(c\) (leaving all other columns or rows of \(A\) unchanged), then \(\det(A')=c\det(A)\). Note: Multilinearity does not imply linearity. For general \(n\times n\) matrices \(A\) and \(B\)  (\(n>1\)) and scalars \(c\in \bfr\), it is NOT true that \(\det(A+B)=\det(A)+\det(B)\), and it is NOT true that \(\det(cA)=c\det(A)\). If \(A' \) is a matrix obtained by interchanging exactly two columns of \(A\), or exactly two rows of \(A\), then \(\det(A')=-\det(A)\). Hence if \(A\) has two identical columns, or two identical rows, then \(\det(A)=-\det(A)\), so \(\det(A)=0\). Combining this with the previous property in purple (in item 8) yields: If one column of \(A\) is a scalar multiple of another column, or one row of \(A\) is a scalar multiple of another row, then \(\det(A)=0\). Since an all-zero column (or rwo) is the scalar 0 times any other column (or row), it also follows that If any column or row of \(A\) is all zero, then \(\det(A)=0\). \(\det(A)\) can be computed by a cofactor expansion along any row or column (paying careful attention to the sign multiplying the first cofactor!) The determinant of a diagonal matrix, or more generally an upper-triangular or lower-triangular matrix, is the product of the diagonal entries   (cf. assigned exercises 4.3/ 9, 19) Elementary row/column operations have simple effects on the determinant: Type-1 operations change the determinant by a sign. The type-2 operations I've denoted "\({\rm Rop}_{2ic}\)" and "\({\rm Cop}_{2ic}\)" change the determinant by a factor of \(c\).   I.e., if \(P\) is either of these operations, then \(\det(P(A))=c\det(A)\) Type-3 operations do not change the determinant at all. For example, if the columns of \(A\) are \(v_1, \dots, v_n\), and \(P\) is the operation I've denoted "\({\rm Cop}_{3ijc}\)", with \(i=1\) and \(j=2\) for concreteness' sake, $$ \det(P(A)) =\det\left( \begin{array} {c|c|c|c} v_1+cv_2 & v_2 & \dots & v_n \end{array} \right) =\det\left( \begin{array} {c|c|c|c} v_1& v_2 & \dots & v_n \end{array} \right) + \det\left( \begin{array} {c|c|c|c} cv_2 & v_2 & \dots & v_n \end{array} \right)\\ = \det(A)+0 =\det(A). $$ For some examples of using elementary row/column operations to simplify the computation of a determinant, see this handout. Determinants and orientation For any nonzero \(c\in{\bf R}\), identify the sign of \(c\) (positive or negative) with the corresponding real number \(+1\) or \(-1\), so that we can write equations involving multiplication by signs, e.g. "\(c={\rm sign}(c)\,|c|\)." Every ordered basis \(\beta\) of \({\bf R}^n\) has a well-defined sign associated with it, called the orientation of \(\beta\), defined as follows: If \(\beta=\{v_1, v_2, \dots, v_n\}\) is an ordered basis of \({\bf R}^n\), where we view elements of \({\bf R}^n\) as column vectors, let \(A_{(\beta)} =\left( \begin{array} {c|c|c|c} v_1 & v_2 & \dots & v_n \end{array} \right) \), the \(n\times n\) matrix whose \(i^{\rm th}\) column is \(v_i\),   \(1\leq i\leq n\). (The notation \(A_{(\beta)}\) is introduced here just for this discussion; it is not permanent or standard.) Then \(A_{(\beta)}\) is invertible, so \(\det(A_{(\beta)})\) is not zero, hence is either positive or negative. We define the orientation of \(\beta\) (denoted \({\mathcal O}(\beta)\) in our textbook) to be \({\rm sign}(\det(A_{(\beta)}))\in \{+1,-1\}.\) Correspondingly, we say that the basis \(\beta\) is positively or negatively oriented. For example, the standard basis of \({\bf R}^n\) is positively oriented (the corresponding matrix \(A_{(\beta)}\) is the identity matrix). With \(\beta\) as above, let \(\beta'=\{-v_1, v_2, v_3, \dots, v_n\}\), the ordered set obtained from \(\beta\) by replacing \(v_1\) with \(-v_1\), leaving the other vectors unchanged. Then \(\beta'\) is also a basis of \({\bf R}^n\), and, by the multilinearity property, \({\mathcal O}(\beta') =-{\mathcal O}(\beta)\). Thus there is a one-to-one correspondence (i.e. a bijection) between the set of positively oriented bases of \({\bf R}^n\) and the set of negatively oriented bases of \({\bf R}^n\). ("Change \(v_1\) to \(-v_1\)" is not the only one-to-one correspondence between these sets of bases. Think of some more.) In this sense, "exactly half" the bases of \({\bf R}^n\) are positively oriented, and "exactly half" are negatively oriented. (A term like "in this sense" is needed here since the phrase "exactly half of an infinite set" has no clear meaning.) If we treat elements of \({\bf R}^n\) as row vectors, and define \(A^{(\beta)}\) to be the matrix whose \(i^{\rm th}\) row is \(v_i\), then \(A^{(\beta)}\) is the transpose of \(A_{(\beta)}\). Hence, because of the general fact "\(\det(A^t)=\det(A)\)," we obtain exactly the same orientation for every basis as we did by treating elements of \({\bf R}^n\) as column vectors. Determinants and volume. (For this topic, some terminology needs to be introduced before the relevant fact can be stated.) (Terminology and temporary notation) For any \(\va_1, \dots, \va_n\in \bfr^n\) the parallelepiped determined by the ordered \(n\)-tuple  \(\a:=(\va_1,\dots, \va_n)\)   (equivalently, the list \(\va_1,\dots, \va_n\)) is the following subset of \(\bfr^n\): $$ \begin{eqnarray*} P_\a &:=&\left\{t_1\,\va_1+ t_2\va_2+ \dots + t_n\va_n \ : \ 0\leq t_i\leq 1,\ \ \ 1\leq i\leq n\right\}\\ && \ \subseteq\ \span(\{\va_1,\dots, \va_n\})\ \subseteq\ \bfr^n\ . \end{eqnarray*} $$ (The notation \(P_\a\) is introduced here just for this discussion; it is not permanent or standard.) Note that if \(n=1\) and \(a=\va_1\), then \(P_\a\) simply the closed interval in \(\bfr\) with endpoints \(0\) and \(a\). For the case \(n=2\), convince yourself that if the list \(\va_1,\va_2\) is linearly independent, then \(P_\a\) is a parallelogram (as depicted in Figure 4.3 [p. 203] of FIS), two of whose adjacent sides are the line segments from the origin to the tips of \(\va_1\) and \(\va_2\). Convince yourself also that if the list \(\va_1, \va_2\) is linearly dependent, then \(P_\a\) is a line segment or a single point (the latter happening only in the extreme case \(\va_1=\va_2= {\bf 0}\)). In the latter two cases we regard \(P_\a\) as a "degenerate" or "collapsed" parallelogram. Although a parallelepiped is not a subspace of \(\bfr^n\), we still have a notion of dimension for parallelepipeds. Specifically, we define the dimension of \(P_\a\), denoted \(\dim(P_\a)\), to be the dimension of \(\span(\{\va_1,\dots, \va_n\})\). If \(\dim(P_\a) < n \)   (equivalently, if the list   \(\va_1, \dots, \va_n\)   is linearly dependent; also equivalently, if \(P_\a\) lies in a subspace of \(\bfr^n\) of dimension less than \(n\)), we say that the parallelepiped is degenerate.   If \(\dim(P_\a)=n\) (equivalently, if the list \(\va_1, \dots, \va_n\) is a linearly independent [hence an ordered basis of \(\bfr^n\)]), we say that the parallelepiped is nondegenerate ("solid''). (More terminology and temporary notation) There is a notion of \(n\)-dimensional (Euclidean) volume in \({\bf R}^n\) (let's just call this "\(n\)-volume") with the property that the \(n~\mbox{-volume}\) of a rectangular box is the product of the \(n\) edge-lengths. The precise definition of \(n\)-volume for more-general subsets of \({\bf R}^n\) would require a very long digression, but for \(n=1, 2\) or 3 the notion of \(n\)-volume coincides, respectively, with length, area, and (3D) volume. For an ordered \(n\)-tuple of vectors in \(\bfr^n\), say \(\alpha=({\bf a}_1, \dots, {\bf a}_n)\), let \(A_{(\alpha)} =\left( \begin{array} {c|c|c|c} {\bf a}_1 & {\bf a}_2 & \dots & {\bf a}_n \end{array} \right) \). (The only difference between this and the "\(A_{(\beta)}\)" used in our discussion of orientation is that we are not requiring the list \(\va_1, \dots, \va_n\) to be linearly independent.) (Fact) For the (possibly degenerate) parallelepiped \(P=P_{(\alpha)}\), the determinant of \(A_{(\alpha)}\) and the \(n\)-volume of \(P_{(\alpha)}\) coincide up to sign. More specifically: If \(\alpha\) is linearly independent, then \(\det(A_{(\alpha)})\ = \ {\mathcal O}(\alpha)\times\) (\(n\)-volume of \(P_{(\alpha)}\)). If \(\alpha\) is linearly dependent, then \(\det(A_{(\alpha)})\ =\ 0\ =\ \) \(n\)-volume of \(P_{(\alpha)}\).
Before the final exam	Assignment 14 Below is some homework related to the post-Thanksgiving material you're responsible for. Its due-date is the day of the final exam. (But the earlier you do it, the better.) Read Section 5.1.        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 exist an invertible matrix \(Q\) and a diagonal matrix \(D\) such that \(A=QDQ^{-1}\). 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. 5.1/ 1, 2, 3abc, 4abd, 5abcdhi, 7–13 , 16, 18, 20.     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 example of less-than-good writing. The sentence should have begun with "[F]or \(n>2\)," not ended with it. In Section 5.2: Read the first two paragraphs. Before reading the statement of Theorem 5.5, read the following easier-to-understand special case: Theorem 5.4\(\frac{1}{2}\) (Nickname: "Eigenvectors to different eigenvalues are linearly independent")    Let  \(T\)  be a linear operator on a vector space. Suppose that  \(v_1, \dots, v_k\)  are eigenvectors of \(T\) corresponding to distinct eigenvalues  \(\l_1, \dots, \l_k\)   respectively. (Remember that "distinct" means   \(\l_i\neq \l_j\)  whenever  \(i\neq j.\)) Then the list  \(\{v_1, v_2, \dots, v_k\}\)  is linearly independent. Although "Theorem 5.4\(\frac{1}{2}\)" is a special case of FIS's Theorem 5.5, and the proof I've given in a handout (see below) occupies more space than the book's proof of the more general theorem, I think you'll find my proof easier to read, comprehend, and reproduce, partly because the notation is much less daunting. Read the handout with the file name "e-vects_to_distinct_e-vals.pdf", posted in Canvas, under Files/miscellaneous notes. The handout has a proof of "Theorem 5.4\(\frac{1}{2}\)", some comments, and two corollaries. The second corollary is exactly FIS Theorem 5.5. Continue reading Section 5.2 up through Example 7 (p. 271). 5.2/ 1, 2abcdef, 3bf, 7, 10.     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. Two non-book problems.

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