Homework assignments and rules for written work
MAS 4105 Section 5287 (24829) — Linear Algebra 1
Spring 2025

Assignments

Date due	Assignment
F 1/17/25	Assignment 0 (just reading, but important to do before the end of Drop/Add) 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 1/21/25	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'll see if you scroll through that page, up through Fall 2023 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 that these notes could be overwhelming. So this year, I'm experimenting with referring you to a different page for many of those notes. 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. Read Section 1.2. Remember that, in this class, whenever the book refers to a general field \(F\), you may mentally substitute \(\bfr\) for \(F\) unless I say otherwise. If an exercise I've assigned refers specifically to the field \({\bf C}\) (plain "C " in FIS) 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 except for parts c and d,  2–4, 8, 12–14, 17–21. Do this non-book problem. Read Section 1.3 through at least Example 3.
T 1/28/25	Assignment 2   1.2/ 1cd Finish reading Section 1.3. Read the current (1/25/2025) version of the handout Polynomials and Polynomial Functions posted on the Miscellaneous Handouts page. (I will be adding some material to this handout over the next few weeks.) 1.3/ 1b–g, 2–10, 12–16, 18, 19, 22.   Note: In 8f, "\(a_1^2\)" means \( (a_1)^2\), etc. for \(a_2^2\) and \(a_3^2\). I did #18 in class very hurriedly on Friday 1/24. I'm having you re-do it in case you weren't able to catch some step(s) in the argument. (It's good practice with the ideas and with proof-writing anyway.) In #22, assume \(F_1= F_2=\bfr\), of course, just as I've said to assume \(F=\bfr\) when you see a single field \(F\) in the book. 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. Exercise 1.3/ 24 There are a few more direct-sum exercises from Section 1.3 that I'll be assigning, but I've moved them to the next assignment to avoid further lengthening the current one. However, they're thematically related to the current assignment, so if you have time, doing these problems now wouldn't be a bad idea.
T 2/4/25	Assignment 3   1.3/ 24–26, 28–30. See additional instructions below about 28 and 30. 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. For #30, you already proved half of the stated "if and only if" as DS3 in the previous assignment, so all that remains is for you to prove the other half. In the handout Lists, linear combinations, and linear independence (posted on the Miscellaneous Handouts page): Before the Wed. 1/29 class, read up through Remark 6. (Definitions 1, 2ab, and 5, as well as Remark 6, corresponds to certain material in Section 1.4 of FIS; Definitions 2, Proposition 3, and Definition 4 correspond to material in Section 1.5. In class, we'll finish covering Section 1.4 before moving on to Section 1.5, but in my notes it seemed advantageous not to split up the material on pp. 1–3 into separation sections.) Also read the "Some additional comments" section at the end of the handout, and the green text in Assignment 3 on my Fall 2024 homework page. (More from this handout added as last item in assignment.) 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, Assignment 2, "Read Section 1.4 ..." bullet-point, 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 than this 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.) Read Section 1.5.       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!). In the "Lists, ..." handout: do Exercise 7, read Proposition 8 and its proof, and read Example 9.
T 2/11/25	Assignment 4   (POSSIBLY NOT COMPLETE YET) 1.5/ 1, 2a–2f, 3–7, 10, 12, 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. 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. In the updated version (2/2/2025) of the Polynomials and Polynomial Functions handout, read this material that's been added since the previous version: The blue note inserted after Proposition 3.1, and the bold-face paragraph at the top of the next page. Everything from Corollary 3.3 through the end of the handout. 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 in a birectional implication "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 if you were in class on Friday 2/7, because we used the exact same argument to prove an equivalent result; we just hadn't introduced the word "basis" yet. It might appear at first that the "unique representation" 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 is neither more nor less general than the unique-representation result we proved in class; the two results are equivalent. In the handout Some Notes on Bases and Dimension (posted on the Miscellaneous Handouts page), read up through Remark 5 on p. 4. Practice writing definitions! Consider this to be a part of every assignment.    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
T 2/25/25	Assignment 6
T 3/4/25	Assignment 7
T 3/11/25	Assignment 8
T 3/25/25	Assignment 9
T 4/1/25	Assignment 10
T 4/8/25	Assignment 11
T 4/15/25	Assignment 12
T 4/22/25	Assignment 13

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