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Honors Linear Algebra Can Be Fun, Cool, Even Beautiful

In this episode of our podcast, David Gardner (CDS’26) talks about his favorite BU class

You can also find this episode on Apple Podcasts, Spotify, Google Podcasts, and other podcast platforms.

The classes we take can change our perspectives and shape our lives—and we think that’s worth celebrating. Our new podcast, Today I Learned, is all about the classes at BU that have had a real effect on students in our community; we want to know all about the classroom environment, professor, subject matter, and the cool facts that make a lasting impression.

For those who think that math is a chore, we present to you David Gardner. In the fifth episode of our BU Today podcast,Today I Learned, Gardner (CDS’26) gets candid about his favorite BU class, MA 442: Honors Level Linear Algebra. If you don’t have a basic understanding of the concept, this might not be the class for you because, as the name suggests, it’s not for the uninitiated. Gardner agrees that every day is a challenge and he’ll often go to Professor David Rohrlich for assistance with difficult proofs, but that the hard work always pays off: “It’s fun,” he says, “it’s a game.” For him, solving each problem is like cracking a code, or completing a puzzle—there’s something purely enjoyable about the experience of finding a solution. What’s more, Rohrlich’s teaching style is attuned toward deductive reasoning and linear thinking, which Gardner says has allowed him to examine nonmathematical dilemmas in history and politics with a new mindset. Be it engineering, technology or, in Gardner’s case, just for fun, this episode will make you see linear algebra like you’ve never seen it before.

Want to be our next guest? Tell us about your favorite class here. Undergraduates and graduate students are welcome to submit.

Takeaways

• According to Gardner, high school math just doesn’t have the kind of magic as college-level math. Whereas the former is more about computing and nailing down basic equations, the latter allows you to apply them in novel ways to solve complex problems.
• Is math art? Gardner says that sometimes the application of a mathematical principle is not necessarily as valuable as the principle itself, and that they have an aesthetic value that should be allowed to stand on its own.
• If you find yourself having dreams about theorems and equations, be sure to write them down when you wake up. You could wind up contributing to important discoveries long after your death.

Transcript

Sophie Yarin: Hello everyone, and welcome to Today I Learned, a BU Today podcast where we explore fun facts and ideas across a variety of academic disciplines by interviewing students about the things they learn in their favorite classes at BU. A single class can have a transformative impact, whether you’re changing majors, picking a career path, or just looking to expand your horizons. I’m your host, Sophie Yarin, and I’m investigating how the things we learn in the classroom affect our lives. To do that, I’m going to be speaking directly with BU students, which is why we have David Gardner joining us in the studio today. David, thank you so much for being here.

Gardner: Thanks for having me.

Yarin: So, you’re a freshman at the Faculty of Computing & Data Sciences, is that right? 

Gardner: That is correct. 

Yarin: And so what’s your major, David?

Gardner: So, my major is currently data science, but I’m exploring math, computer science, and possibly mechanical engineering. So, right now I’m taking classes in those colleges, kind of trying to decide which path I want to take, which is why I’m taking MA 442, which is not actually in the data science curriculum, but I can take the class instead of taking one of the other data science foundation classes.

Yarin: Got it, keeping your options open. 

Gardner: Yeah.

Yarin: You’re a freshman, you can do that. So, the course that you’re going to be talking about with us today is MA 442: Honors [Level] Linear Algebra, with Professor David Rohrlich. Before we get into a fun fact, which we typically do, I just wanted to sort of nail down a brief overview for our less mathematically inclined listeners like myself. What’s algebra?

Gardner: That’s a great question. So, in high school, when you take algebra classes, it’s computation, mostly. You have a formula and then you have to apply it. But in college and upper-level mathematics, algebra has to do with number systems—what is addition, what is subtraction, why is that important, and also, that you expand beyond the real numbers. You may talk about, like, imaginary numbers, or you’ll talk about rings and vector spaces, and all these really technical terms to provide definitions so that you can create these other theorems, and then build off of those convenient definitions. And then for linear algebra—linear algebra is basically algebra, but all the nice, useful things from algebra that can be applied to computer science, or engineering. And that’s why it’s required for engineering students, computer science students, and data science students, as well.

Yarin: Okay, so if I understand you correctly, we’re taking these fundamental computations, like addition and subtraction, and sort of adding extra elements and dimensions to the formulas that they’re in?

Gardner: Yeah, so it’s interesting, because the first day of class, my professor, Professor Rohrlich, he gave us a definition of a vector space, which I won’t go into, but basically the very boiled-down definition is: you have some numbers, and you can multiply them together, and you can add them together. And that’s what a vector space is. And then we build off of that, we get into functions, matrices, which you’ve probably heard of if you took any kind of high school math. And those are really useful—matrices are very useful for computers. A lot of computations for computers are actually just giant matrix multiplications and matrix operations.

Yarin: Okay, so some of the familiar cast of characters coming back in new ways, like matrices and linear equations. All right, that’s a good foundation, I think, for us to build off of. So, now let’s dig into some specifics, shall we? I want to start by asking you to share something cool that you learned in class.

Gardner: So, one of my friends, he brought up Ramanujan, who was this 19th-century mathematician who came up with lots of theorems in his dreams.

Gardner: Yes, I was so shocked. And he was able to approximate like, whatever, pi in 17 different ways. And if you’ve looked them up, if you’ve looked up the approximations, they look really, really crazy and really random. And they still work.

Yarin: So, the stuff that he was dreaming of actually turned out to be proven as mathematically sound?

Gardner: Yeah, so one example of that is that he had some theorems that Stephen Hawking used to look at, or to explore the properties of black holes. And Ramanujan didn’t know that black holes existed when he was alive.

Yarin: Wow. So, kind of like a mystic, almost.

Gardner: Yeah.

Yarin: That’s really cool.

Gardner: Math is crazy.

Yarin: Math is crazy! All right, let’s keep going. So, where do we see some of these concepts emerging organically outside of the lab or the classroom?

Gardner: It’s really interesting, because I’ve seen a lot of crossover between my data science class—I’m taking DS 210, which is [Programming for Data Science]—I’ve seen some crossover between that and linear algebra. For example, we learned about interpolation, which is basically if you have a bunch of points on a graph, and you want to fit a function, so it goes through every single point. That’s what interpolation is. And in linear algebra, I learned about Lagrangian interpolation, which is basically interpolation, but we proved how it works and why it was true, and that was a cool crossover. And Lagrangian interpolation, if you’ve taken Calc II, you’ve definitely done partial fraction decomposition when you’re evaluating an integral, and that has to do with interpolation. So, as I just said, there’s a lot of crossover for the math or STEM people out there. If you’re not into STEM, linear algebra has applications in cryptography, because it can be used to encode messages and decode messages, if you have the key or you have this matrix as the key to figure out what the message is.

Yarin: Okay, so it sounds like there are a lot of applications within and around STEM, but I know that when you and I had talked briefly before we started recording, you had mentioned that this class has also influenced the way that you approach political and scientific arguments. So, I would love to hear more about that.

Gardner: Yeah, so what’s really unique about this math class is that it’s the first real math class I’d say I’ve taken, because with all the math classes I’ve taken before, it’s more focused on, again, computation. So, you’re given some formula or some idea, then on the tests, you have to apply those formulas. But instead, in Rohrlich’s class, we have to prove all the theorems we learn and he goes through the proofs with us. But we have to understand why the theorems are true and actually on the test sometimes he’ll put these incomplete proofs, and we have to say why the one step of the proof is true and give reasoning to why it’s true instead of doing all this computation. So, the way that applies to politics, or like reading some newspaper article, is that, like, now I have a stronger sense of deductive reasoning. And because I’m forced to question whether everything is true, and look at every single step, I’m more aware of catching things that aren’t true, catching, like, something that doesn’t seem right or questioning whether my beliefs are true or not.

Yarin: Right, and I hear where that sort of deductive logic comes in, where everything has to flow into the next thing in order for you to understand the whole picture, it sounds like. So, just switching gears here for a second, what are you looking at, career-wise?

Gardner: Yeah, this is a little unrelated to the class, but I’m interested in environmentalism. So, I’m in BU’s environmental fraternity, which is called Epsilon Eta, you should definitely check them out. And I want to do something in tech and environmentalism. So, I’m not really sure at the moment. I’m interested in being in climate tech, which is what it sounds like—it’s environmentalism and technology coming together.

Yarin: You had mentioned earlier that this linear algebra is something that you need to understand in order to understand how computers work. So, you would need this for tech, right? As a prereq?

Gardner: Yes, linear algebra, as a class, it’s very useful. But you go beyond that, and it starts getting…

Yarin: A little niche? 

Gardner: Like, not many applications. It’s more aesthetic, it’s not really applied.

Yarin: When we had talked earlier, you had talked about the play that comes with drafting and finalizing proofs. And it’s really interesting that you mentioned play, like a sense of play. And I’d love it if you could expand on that.

Gardner: I think math is really fun. I know a lot of people don’t like math, and I understand why because high school math is not real math, it’s mostly computation. And it’s not the fun part of math. So, the fun part of math is you’re given a problem, you’re given some result you’re trying to find, and the fun part is figuring out the reasoning to get to that point; it feels like a game. It’s more that you’re solving a puzzle, not really just filling out this predetermined path to take, if that makes sense. Yeah, what I like about the homework is that it’s like a new game, for me. I look at the problems and, at first, I might be like, really confused. I have to think about why that’s actually true, or figure out how to get there. I’ll spend lots of time just thinking.

Yarin: So, David, do you believe in math for math’s sake? Does it need to have a function in a larger structure like tech? 

Gardner: Yeah, I don’t really know where I stand on that, but one of my good friends in that class, he told me that he really doesn’t want his mathematical studies to have any real-world application. He sees math as an art form. It’s purely aesthetic. So, at first, I was really confused why he was saying that, but then I realized, like, that does make sense, because math is very beautiful. You build math up from these very simple definitions, and there’s these crazy results. And it’s all based on, like always, it’s all based on reasoning. It’s like you’re pulling something out of a hat. There’s reasons why everything is true. It’s just really beautiful to see the results of that. But also, that same friend was saying, well, he also studied number theory, and he was saying how he’s mad that computer scientists have made number theory useful. 

Yarin: Oh no, shots fired. 

Gardner: But I think that’s really funny. And he’s obviously joking, but also like, it’s really cool that these math concepts are reused in computers to improve everyone’s life. Because we have computers everywhere, and they’re super efficient, they store so much information. And without math, I don’t think that could have existed.

Yarin: So, if you wanted to recommend this class to a friend, and maybe a friend who is good at math but is not necessarily passionate about math, how would you describe it?

Gardner: I’d say it’s fun. It’s a game. Yeah, like, it’s a difficult class, you have to dedicate a lot of time outside of class, but the time I’m spending is not boring or annoying. It’s usually entertaining for me. And I’d say that your definition, or your concept of what math is, is probably wrong. But if you like math, if you like that kind of reasoning, then I think you’ll love this class because of all the logical reasoning that’s involved.

Yarin: So, take this class to change your mind about math.

Gardner: Yes, but also I know there’s a lot of people who hate math out there, so I would take the class for a week or two and if you hate it, drop it, because you’ll definitely not do well. If you don’t like what you’re doing in the first few weeks.

Yarin: That’s good, we love some honest candidness on the show. So, David, thank you so much for sitting down with us and helping us put two and two together. I can honestly say this is the most I’ve thought about math since high school and I think I speak for our more right-brained listeners out there when I say I found it really enjoyable. I really liked it. So, folks, thanks for tuning into Today I Learned, a BU Today podcast. Do you have a favorite class you think we should know about? Tell us all about it by filling out the form linked in our description. Today I Learned is produced and engineered by Andrew Hallock and edited and hosted by Sophie Yarin—that’s me. We’ll see you next time. Stay curious out there! 

Editors Note: In the audio above and in an earlier version of this text transcript, Gardner referenced “vector fields” when he meant to say “vector spaces”. The text has since been updated.