Hi [redacted], 

 
The set of all 2 x 2 matrices forms a vector space, with the usual notions of matrix addition and scalar multiplication. You can check this yourself if you want; just go through that list of 10 vector space axioms and verify them all. The point here is that when you are defining a vector space, you are free to make your vectors whatever you want. If they happen to have some other interpretation in linear algebra as well—as is the case here—so be it. You don’t necessarily have to have any deference to that other interpretation. You are the vector space creator, and you get to say what a vector is. Here, your vector space elements are matrices. 
 
So then the question is what is the dimension of this vector space of 2 x 2 matrices. Any matrix [a b; c d] can be written as 
 
a [1 0; 0 0] + b [0 1; 0 0] + c [0 0; 1 0] + d [0 0; 0 1] 
 
so clearly those four matrices span the space. Furthermore, if that linear combination above is equal to the zero matrix (which is the zero vector of this vector space), then all four weights have to be zero. So it’s also linearly independent. So it’s a basis! 
 
What I mentioned in class is that this vector space is really just R^4 in disguise. Basically you just write the last two components to the side instead of beneath the first two, and you get the same thing. In precise language, the function from M_{2,2} to R^4 which sends [a b; c d] to [a; c; b; d] is a vector space isomorphism. You can check that from the definition if you want, but if you think about it for a bit, it should be clear that everything (meaning vector addition and scalar multiplication) works the same way in the two spaces, except for how you arrange the entries. 
 
SW
 

On Apr 10, 2017, at 1:15 AM, [redacted] wrote:
Hi Professor Watson,

My name is [redacted] and I’m in your Linear section. In the review session, I was a little confused about the problem what we did in which a 2x2 matrix that was subject to some transformation. I was just wondering why in this case this represented 4 dimensional space, rather than 2 dimensional space based on the number of columns?

Best,
[redacted]
08 April 2017

Dear class, 

 
We will have an exam review session tomorrow (Sunday) evening from 19:00 to 21:00 in Barus & Holley 168. There is also a review session hosted by Isaac on Monday evening from 5 PM to 7 PM in Wilson 102
 
Other announcements: the practice midterm solutions are available. The very last question—about showing that the intersection of those two vector spaces has dimension at least 2—was a little on the hard side. If you feel you’re not on top of the material at the moment, I wouldn’t worry too much about that one. 
 
Finally, I wrote a Quora answer yesterday essentially on “what’s the deal with abstract vector spaces?” I wrote it with the intention of sharing it with the class, because several folks have asked that question one-on-one and it felt worth elaborating.  
 
SW
 
P.S. Info rundown: 
 
Sections covered: 2.2, 2.3, 2.8, 2.9, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6
Exam Location: Salomon de Ciccio Auditorium
Exam Time: Tuesday evening from 6 PM to 8 PM. 
Review Sessions: Sunday and Monday evening
Resources: practice midtermextra practice problems with solutionsguide to working your way through proof problemscheatsheet

04/06/2017

Dear students, 

As you know, we have a midterm next Tuesday from 18:00 to 20:00 in Salomon de Ciccio Auditorium. For students who cannot take the exam during that time, please fill out this form so that we can arrange separate accommodations. 
 
For students who have SEAS letters, I will use the list from the first midterm and get in touch with you separately. If you have a SEAS letter and were not on that list from the first midterm for some reason, send me an email to let me know. 
 
Just to recap for everyone: special resources for this midterm include a set theory review, a list of practice problems with solutions, some advice for solving the kinds of problems we’ve been doing lately, and of course a practice midterm (solutions forthcoming). 
 
Best, 
SW 

 

04/06/2017

Dear class, 

I learned in office hours yesterday and last night that some folks are having a difficult time with writing up solutions to problems which involve relating various abstract vector space notions (linear independence, linear transformations subspaces, etc.). Most of these difficulties are, I think, not particularly related to linear algebra and are quite surmountable. I have written a document which contains the advice I shared with the students who were at office hours last night. It provides a detailed explanation of what kinds of thought processes help you get unstuck with these kinds of problems, in the context of one particular example. I hope that you find it useful. 
 
Best, 
SW 

 

04/04/2017

Dear students, 

For your convenience (and mine), we will be holding office hours today 12:00 to 2:00 in Watson CIT Building Room 227. 
 
I will also be in my office from 17:10 or so (maybe a bit earlier) to 18:00 this evening working with at least one student on proofs. Others are welcome to join. You’ll want to get there before 17:00, though, because the doors to the building lock at that time. If you arrive a bit late, you can take your chances with finding someone walking around the first floor who will let you in. 
 
Also, the practice midterm is up
 
SW

 

03/24/2017

Dear students, 

 
TL;DR (1) practice problems for Midterm II, (2) sections covered on Midterm II, and (3) videos from lecture. 
 
Realizing that many of you felt that there was not sufficient time dedicated to examples during lecture, I have uploaded a set of 10 worked problems (solutions here) designed to help you develop greater facility with concepts most centrally featured on the upcoming exam: linear independence, span, subspaces, and linear transformations in the context of abstract vector spaces. Please think of this resource like a pre-practice midterm, as the difficulty level isn’t necessarily tuned to the that of the exam, the scope is narrower, and the formatting is very different. There is an actual practice midterm forthcoming. 
 
On the topic of exam resources, I would again like to direct your attention to the summary of learning objectives. This list is helpful for identifying examinable topics (the exam will cover weeks 5 through 9 in that list; in the book: 2.2, 2.3, 2.8, 2.9, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6), but it is also helpful for consolidating knowledge. The bullet points help you distill important ideas into memorable phrases, which is important for building up a toolkit of ideas and facts you can apply when problem solving. 
 
Other resources include lecture videos linked from the “class notes and videos" section of the course website, which I have begun uploading in fulfillment of a course survey request. These are offered as-is, since they are not intended as a substitute for class attendance (except the ones from the 14th of March; that was the snow day, and those videos are supposed to be of a higher production quality). 
 
Have a nice spring break! 
 
SW

03/22/2017

Dear student, 

Because I made HW7 from HW6 by adding 7 days to the due date and changing the question, last week’s due-date typo was carried forward to this week. Although I hope that everyone understood the intended deadline (which has always been end of the day Wednesday), I can’t rule out the possibility that some of you might have genuinely misread the deadline and planned accordingly. So I feel I have no choice but to change the deadline to end of the day Thursday
 
I apologize for the inconvenience, such as it is, and I can assure you that the deadline for HW 8 is printed correctly. 
 
Best, 
SW
 

03/20/2017

Dear student, 

If you have found that you are a little uncomfortable with some of the recent set- and function-related vocabulary and notation we’ve been using in class recently, you might want to check out a brief discussion I added to the course documents section of the course website. I have attempted to include everything necessary to get you up to speed the set theory ideas we use in this class. 
 
Let me know if you have any questions. 
 
SW

 

03/16/2017

 

Dear all, 

 

One student pointed out that I mistyped the deadline for the homework that was due last night, and as a result I have re-opened the assignment in Gradescope. You can resubmit if you want to, or if you didn’t submit the homework, you have through tonight (until 04:00 tomorrow morning). I suspect this won’t affect most of you, but you deserve to know. 

 

Also, thanks again to everyone who made helpful suggestions in the course survey. I will recap some of these and respond to them below the fold, for those who are interested. There is no expectation that you read them, although if you made a fairly specific remark, there’s a decent chance you’ll find a reply below. 

 

SW 

 

———————————————————

 

1. MyMathLab homework too long. This is easy: I can keep the number of problems capped at 30. For those who preferred more practice, bear in mind that you can practice as many MML problems as you want, even after you’re done with your homework. The system gives me an expected amount of time the homework I assign is predicted to take, and it has never been over 75 minutes (!). I was beginning to realize that these numbers were not realistic, but they seem to be substantially so for much of the class. So I can adjust. 

 

I would add that if the homework takes you much longer than you feel it should, then you could consider doing more preparation before starting the homework. It might be more efficient to read the appropriate section in the book or in the course notes to have the relevant concepts loaded up in your mind before you begin the homework. 

 

2. Questions during class. I agree that the interactivity during class needs improvement, especially for the students who may feel hesitant to ask questions in front of a large class. I’m interested in piloting a chatroom/classroom hybrid idea, where people can submit questions anonymously using their smartphones or laptops. This would involve dedicating slightly longer periods to questions, to give people a chance to type. Today in class we also tried turn-and-talk. 

 

3. Pacing during class. Several folks expressed concern that I linger too long on certain ideas, often focusing on explaining mental models or analogies at the expense of detailing certain important procedures or having time to fully cover all topics. I recognizing my tendency to do this, and I will work on tightening the presentation, selecting illustrations and reflections more judiciously. 

 

4. This class is way too easy/hard. These kinds of statements were all over the map, which makes them difficult to accommodate simultaneously. I will say, though, that quite a few students are concerned about the course more difficult. I understand this; you made decisions about how to schedule your semester based on a certain workload expectation, and it can be quite disruptive if that changes. Here are some thoughts: (a) one respondent suggested optional challenge problems for the more advanced students. I think this is a good idea, and I can at least produce a short list of such questions from the book each week. And (b) the material is naturally going to get more interesting anyway as we get more concepts under our belt. I will try to compensate for any increased difficulty with other improvements (clearer lectures, shorter MML, etc.), so the workload remains moderate. 

 

5. Because the exam scores were so high, a few silly mistakes cost me an A. I agree that this is not great. I would prefer the exam experience to be very challenging without feeling punitive. Ideally you could demonstrate your knowledge in a way that is as insensitive as possible to mistakes that don’t reflect your understanding. Note that I tried to do this on the first exam by avoiding arithmetically intensive problems. Also, I think the next exams are going to be better in this regard, for the simple reason that setting problems which are both fair and challenging is difficult when most of the content boils down to row reducing some matrix and then properly interpreting the result. As we move away from everything being a matrix, this will get easier. 

s. 

 

6. It’s wasteful to start each problem on a new piece of paper. You don’t have to do this, really. What’s important is that your submissions are distinct. So you can write your solution on the same piece of paper and then crop your scans down appropriately. Or, if you are so inclined, you could write all your solutions on a tablet and export to PDF, saving all the paper. Or you could type up your solutions and export to PDF that way. In any case, this policy actually makes a really big difference to the graders, because the additional task of figuring out where on the page your solution is quite distracting. The reason this issue doesn’t apply to paper-graded assessments is that those are typically graded per student for practical reasons. We grade your work per problem, which is more efficient and leads to greater fairness and transparency for you. 

 

7. Let’s have separate homework deadlines for GS and MML. I asked about this in class today, and it was shot down with great fervor. 

 

 

03/13/2017

 

Dear students, 

 

TL;DR for Watson’s section: watch videos and meet for class virtually tomorrow.  

TL;DR for other sections: office hours tomorrow canceled, instead tonight at mathim.com > chatroom math520.  

 

Because tomorrow’s class is canceled, and especially since this is our second weather cancellation of the semester, we need to figure out a remote working arrangement that allows us to keep pace with the other sections. This plan has two main parts: 

 

1. I have uploaded four video lectures for you to watch. I made them today, and they are pretty much exactly what I planned to do in class, except they’re more concise. The notes have a little more detailed writing, and the videos have more detailed spoken explanations, so I recommend reading/watching both, perhaps simultaneously.  

 

2. Obviously you can't ask those videos questions, so it isn’t quite the same as class. Tomorrow from 14:30 to 16:00 I will be available at mathim.com  in the chatroom named math520. It’s difficult to overstate how incredibly low-hassle this is on your end: open a browser, type mathim.com, type math520 into the “chat room name” box, select “Starting chatting!”, and type in whatever nickname you want. That’s it. The chatroom is math equipped, so we can typeset math stuff in the chatroom, or you can just ask questions in plain English and do your best to work around math notation issues. You can choose an anonymous handle, so there’s no reason to feel self-conscious. 

 

3. Since office hours are canceled for tomorrow, I will have a chatroom office hour tonight from 20:00 to 21:00. Just visit mathim.com and type math520, exactly like the directions in #2. If it seems helpful, I might have another such office hour before the Wednesday homework deadline. 

 

Let me know if you have any questions. 

 

SW

 

03/12/2017

 

Dear students,

 

TL;DR please scan or app-scan—don’t photograph—your Gradescope submissions. Also, survey

 

Longer version: 

 

1. I am in the process of figuring out ways to make the Gradescope grading work better for everyone. For the graders, as I have mentioned, I have emphasized the message that we must turn the homework around in a more timely fashion. This is important, and we will get it right. 

 

Conversely, for the benefit of the graders, it is important that homework submissions be handled in a way that makes them easy to read. The Gradescope instructions specify that your submission be processed either using a scanning app (see here for iOS and Android recommendations), or using an actual scanner. Plain photos are much less readable, because the scanning step introduces a contrast between the background and the text. Furthermore, I doubt you will find that using the scanning app is burdensome; in my experience, digitizing documents using Scannable is actually easier than taking photos. In any case, I have granted permission to the graders to include this in their grading rubrics. If there is some reason that you are unable to conveniently and satisfactorily get your homework into Gradescope, let me know and I will help you figure out a solution. 

 

2. Thanks for those who have already responded to our course survey  I have read all the submissions we’ve received so far, and there are some key common threads as well as some excellent suggestions. I will send a follow-up email exploring some of these in more detail, but for those who haven’t participated yet, I’ll leave the survey open until Tuesday evening. 

 

SW

 

03/09/2017

 

Dear students, 

 

1. Everyone’s exam should be available to view in Gradescope. Those who took the exam at an alternate time will find their results listed under a separate assignment title. 

 

2. Please take a few moments to fill out this brief anonymous survey on your experiences so far in Math 520. This is a good opportunity for you to let us know what is working for you and what sort of changes you’d like to see. 

 

3. Related to #2, I recognize that the Gradescope homework turnaround time has not been good so far. I have taken various measures to work toward getting that process running on a regular schedule (one week is the goal), and I am hopeful that the latest batch of changes will get us on track. 

 

Best, 

SW

 

02/27/2017

 

Dear students, 

 

Because this is a midterm week and my office is rather small, we will have office hours in a non-office location: Watson CIT Building Room 227. 

 

Also, at risk of repetition: you have a midterm this week. Furthermore, are no drop grades for midterms. If you have not arranged to take the exam at an alternate time, you are responsible for taking the exam this Thursday from 18:00 to 20:00. 

 

Finally, there are two review sessions, open to all students. Both are in Metcalf Auditorium from 20:00 to 22:00. One is on Tuesday night (IS) and one is Wednesday night (SW). 

 

SW

 

02/25/2017

 

Dear class, 

 

0. Reminder: if you have a schedule conflict or a SEAS accommodation, please submit this form today. 

 

1. There has been some confusion regarding the sections covered on our first midterm, and I apologize for that. As I indicated in my previous email about the midterm—though I didn’t draw enough attention to it—there is a week-by-week

 

!!! —>   SUMMARY OF LEARNING OBJECTIVES  <— !!! 

 

available on the course website under the “Course Notes” section. The exam will cover WEEKS 1–4 on that sheet. 

 

2. Midterm locations are determined alphabetically: 

 

Abele through Ishikuza: Barus & Holley 166

Jackson through Nakai: Barus & Holley 168

Nam through Zhuk: Metcalf Auditorium 

 

3. We are currently planning review sessions for Tuesday night (IS) and Wednesday night (SW) from 19:00 to 21:00. Locations will be announced Monday. 

 

Let me know if you have any questions. 

 

SW

 

02/23/2017

 

Dear students, 

 

If you are able to take the Math 520 exam at the scheduled time (next Thursday from 18:00 to 20:00), please disregard this message

 

A number of students have emailed their instructors to indicate that they have schedule conflicts or extra-time accommodations (approved through SEAS) for the first midterm, which is next Thursday, March 2 from 18:00 to 20:00. We will coordinate such conflicts and accommodations using Google Forms, and in two phases. First, use THIS FORM to indicate your conflict or SEAS accommodation status (in the latter case, you will have already given your instructor a letter from SEAS, but please fill out this Google Form anyway—the point is to make sure we have everyone properly accounted for). We will follow up with the respondents to coordinate further details. 

 

I would urge you to please attend the main administration of the exam, and only submit the above form if you have a genuine schedule conflict or a SEAS letter. If you do submit the form, please do so by SATURDAY if possible, so that we can ensure you are included in all subsequent communications. 

 

Best, 

SW

 

02/19/2017

 

Dear class, 

 

1. For this week only, the deadline for both homework components will be end of day Thursday instead of Wednesday. 

 

2. I will have office hours 12:00 to 14:00 on Thursday instead of Tuesday, since Tuesday is a holiday. 

 

3. There is a practice midterm available (also linked from the course site). To clarify: the practice midterm is not an exhaustive list of the topics you need to know (but this is; weeks 1-4 only). The approach I recommend is to prepare for the exam first. Then take the practice midterm a few days before the midterm date (with simulated exam conditions), and use your performance to gauge how much more studying you need to do and/or which topics to focus on. A good alternative is to work through the problems like a homework assignment over the next week or so. In keeping with this philosophy, I will provide solutions a few days before the midterm. This will give you an opportunity to grapple with the problems without the temptation to look quickly to the solutions. 

 

SW

 

P.S. Vis a vis visualizing linear transformations; this one (h/t Alexi Frongillo) is pretty and runs in your browser as soon as you click the link (just drag that slider on the bottom to see where the gridlines go). 

 

02/15/2017

 

Dear students, 

 

As I announced in class today, some students drew my attention to the fact that Question 4 on the present written homework (HW#3) is somewhat confusingly worded. I apologize. The “system of equations” whose solution set is the span of the given vectors is in fact just one equation. The phrasing “write down a system of equations whose solution set is equal to the span” is kind of misleading, at least to the extent that it makes it sound like no calculation is involved. We should apply the technique on page 4 in these notes, and that does involve some work. I will try to make this up to you in some part by instructing the graders to approach this one very leniently. 

 

Also, I added a function to our course Julia package which allows you to input an arbitrary 2 x 2 matrix and see a movie representing the corresponding linear transformation, with a slider. I put an example in the tutorial notebook (at the end, although you’ll need to actually run the code to get the slider). You’ll want to run Pkg.update() if you have already installed the package—otherwise, you can copy-paste the few lines of installation code from the aforementioned notebook to get up and running. 

 

Best, 

SW 

 

02/07/2017

 

Dear class, 

 

1. This is your second and final homework reminder: tomorrow (Wednesday) night, end of day. After this week I’ll be leaving it to you to remember that homework is due every Wednesday night. 

 

2. Today after class, a random person who was in the room for some event happening next heard me talking about the definition of span with a student and recommended 3blue1brown. I thought whatever, might as well check it out. I’m glad I did: these short YouTube videos are made with some really excellent animation software and illustrate some of the concepts I have trouble conveying vividly with arm motions and by-hand sketches. The narration is tightly choreographed and reflects a deep understanding of what makes linear algebra cool and what ideas are really important. I’m tempted to list their conciseness as another pro, but they’re so good you might decide to push aside House of Cards in your TV-watching routine. 

 

TL;DR I added a new link to a video series to the course resource list, I recommend it highly to everyone. Very easily worth your time, especially if you want clearer geometric intuition about vectors and span

 

SW

 

P.S. For those using the course Julia package, I made some updates. You’ll want to put Pkg.update(“IntroLinearAlgebra”) or just Pkg.update() into a cell and do shift+enter to get them. 

 

02/06/2017

 

Dear class, 

 

For those wholly uninvolved with our ongoing effort to accommodate student scheduling conflicts, you may disregard this message. 

 

The course staff has sent out all the override requests we have been authorized to grant. We tried to do this as fairly as possible, and we are grateful for your honesty in expressing the full range of "desperation scores" from 1 to 5. If you gave a low score and didn't get a spot, hopefully your role in ensuring that that spot went to someone who needed it more urgently than you provides some solace. Please know that we did everything we could to accommodate everyone, including badgering the Registrar pretty much constantly for a larger room that never opened up for us. 

 

TL;DR The dust has pretty well settled at this point. If you want to discuss this further, the best option would be to come by during my office hours tomorrow (noon to 2, in Kassar 312). There isn't really anything I can do as far as granting further overrides, but I can discuss your situation with you anyway and perhaps brainstorm other resolutions. 

 

Best, 

SW

 

02/02/2017

 

Dear class, 

 

For those still looking to change sections, please complete the override request form (also linked from the top of the main course page). We ask that you submit these requests by Sunday.  

 

Best, 

SW 

 

02/01/2017

 

Dear students, 

 

This is a reminder that there is homework due at the end of the day today (technically 4:00 AM tomorrow morning). This includes written problems which you will submit through Gradescope, as well as online homework you’ll complete using MyMathLab. You can obtain temporary free access to MyMathLab, so you should still submit this week’s homework even if you are planning to wait a bit longer to pay for the software. I will not be cluttering your inbox with such reminders throughout the semester, but it seems appropriate for the first week or two for the benefit of students who might have joined the class late. 

 

I will also mention that I’ve added a reduced row echelon function (rref) as well as some help documentation to the course Julia package. My hope is that those who choose to use this resource will do so to (1) relieve some of the burden of row reducing, once by-hand mastery has been achieved, and (2) help to ensure accuracy and efficiently locate arithmetic errors when practicing by-hand calculation. Feedback welcome, as always. 

 

We’re currently working on getting a bigger room for one of the sections, and we are hoping that will resolve many of the issues where folks need to get into sections that are currently full. We will post a Google Form on the course website later this week (probably by Friday, depending on the status of our room search) and use the results of that survey to grant overrides starting early next week. 

 

Yours, 

SW

 

01/27/2017

 

1. If you are hoping to get into one of the sections that is currently full, the department has asked that we see how it goes for the first week or so before taking steps to accommodate particular cases. You are welcome to email me in the meantime to let me know your situation, but I will more helpful starting middle to late next week. And if you have already emailed me, please email me again at that time. 

 

2. I received a helpful reply yesterday from a student who found a way to save money on their access code. I will pass along their advice for those who might be interested: 

 

I purchased my access code last night on eBay, and it worked as expected---here are some examples 12, but I encourage students to search on their own. Sellers tend to send the access code via eBay message or email once payment is cleared. Of course, there are likely cheaper ways to purchase the access code, but as always, one should be cautious when they purchase from any third party seller, and to check out seller reviews before purchase; my personal rule is to not purchase from sellers with minimal (< 200 reviews) or negative (< 98.5%) feedback on eBay.

 

 

01/27/2017

 

Dear Linear Algebra students, 

 

Welcome to Math 520! For those Sections 01, 03, 04, and 05: I am the course head, which means that you will sometimes receive emails from me in addition to the emails you receive from your section instructor. Here’s the main course website in case you missed it; it’s accessible via sswatson.com as well. 

 

Since I received many messages from students about Section 2 (the Tuesday-Thursday 2:30 PM one), I should let you know that the section’s enrollment cap has increased by 99. So if there are students currently in other sections who preferred the Tuesday-Thursday schedule, there should be no problem switching. The new classroom is listed as the Metcalf Research Building Auditorium. 

 

I would like to provide some important information about MyMathLab which I did not mention in class: it is possible to gain access to MyMathLab for free for the first 14 days (see this FAQ, near the bottom). This may be very helpful to you if (i) you aren’t sure you’re going to stay in the course forever (although you can request a refund within the first 15 days), or (ii) you want to buy the book+MML combination, and the book store doesn’t have them in stock yet. 

 

The office hours of all the instructors will be finalized within the next week or so. Our policy will be that any student may attend any instructor’s office hours. 

 

----------------------------------------------------------------------------------

[Inessential section warning] 

 

Some students are justifiably lamenting the cost of the MyMathLab access code. I philosophically prefer free materials, and I always look for them first. I am loathe to require students to shell out money. So I feel it’s appropriate for me to justify the decision. 

 

Experience indicates that timely feedback adds a lot of value to the student learning experience, and traditional paper-based grading is massively outmatched by online homework platforms in this regard. Another major student need addressed by MyMathLab access is access to additional practice problems beyond what is assigned in homework (and specifically problems for which feedback is available). 

 

Here’s one way to think of it: you’re already putting in at least $1800 of your own time (assuming you value it at $10 per hour) to learn this material, not to mention whatever tuition comes out to. So the percent increase in total cost is marginal, and if the improvement is substantial it will easily be worth it. 

 

That said, I am sensitive to the fact that money is not easy to come by for everyone, even in situations where the thing you need to buy may be “worth it”. My understanding is that financial aid office is able to check whether you qualify for assistance (from them) with this sort of course expense, and I also understand that they are very open and helpful. If you need more information about that, I can help you track it down (although I think it’s as simple as googling ‘brown financial aid’ and reaching out to them). 

 

[end of inessential section]

----------------------------------------------------------------------------------

 

As always, email me if you have any questions. 

 

Yours, 

SW 

 

P.S. I boldface keywords in emails because at some point I realized that many people prefer the cliffsnotes version.