jumpingfish

From the top of the ridge, sitting in an alpine meadow. With a blue sky overhead and a cooling breeze blowing. With mountains on the horizon and a desert between. White Sands shimmered in the distance.

I played a video for them. They were sitting in their desks and standing in the back of the room and sitting on the floor in the front.

It was a time lapse video of the northern lights. I played it to show them the wheeling stars and beauty of a cold winterscape and to watch the dancing auroral lights. I played it to talk about numbers.

The music played. Ribbons of green danced in northern skies.

“If you were a geologist,” I said, “you could use numbers to explain why those mountains in the distance are flat.”

“If you were a meteorologist,” I said, “you could use numbers to explain why those clouds are shaped like that.”

“If you were an air traffic controller,” I said just after a time-lapse jet streak flashed, “you would use numbers to explain why that jet was following that particular path.”

“If you were an astronomer, you could use numbers to explain why the stars seem to be turning in circles in the sky.”

“If you were a botanist, you could use numbers to show why those pine trees still have their needles but those other trees have dropped their leaves.”

We talked about a lot of things that morning. We only had an hour or so, but we covered a lot of ground. I told them that I was there to talk about why numbers are such a big part of our lives, whether we’re scientists or engineers or artists or musicians or politicians.

And I told them I’d be back.

Friday I’m going back. We’ll be timing our heartbeats. 

We’re standing there in the parking lot. Standing in a circle waiting for the rest of the group to arrive. Waiting for our workout to begin.

There was a loud bang and some oblong green thing the size of the tip of your thumb bounced off the trunk of a car and rolled on the pavement at our feet.

“Oh,” someone said, “there are so many of those … those green seedy things (if that’s what they are).”

I was at a loss for words. I still am. She’s fast — much faster than I. She’s friendly — always offers encouragement as she passes by on our workouts. But really. She doesn’t know what an acorn is.

On the other hand…

I am reminded of a time long ago when I heard someone describing the final exam for a class she was taking at UT. We were all engineers, but she was taking botany to broaden her horizons. And she was telling us that for the test, the professor walked each of them around campus and had name the trees and plants by their botanical names.

I was stunned. How could anyone do that? Anyone but a botanist. Quercus. Berberis. Asclepias. Ulmus. Helianthus. How could any mere mortal be expected to…

But… acorn.

What a cranky snob I am.

Welcome to E 501. I’m happy to be here this morning; I hope you all are, too.

Just to make sure… this is E 501, Advanced Linear Algebra for Engineers. If you’re in the wrong place feel free to leave now and find the classroom you’re looking for. You’re welcome to stick around, of course, but if you need to go somewhere else, I won’t be offended.

…

I have a confession. Once when I gave that welcoming speech, every single person in the room got up and left. I thought it was some kind of joke. It turned out that I was the one in the wrong place. So I’m glad to see we’re all still here.

So… Let’s get started.

This is Advanced Linear Algebra for Engineers. Before we dive in, let me explain what that name means and give you a sense of how I think about this course.

I’m sure most if not all of you have studied linear algebra before. You probably had some in high school. You’ve worked with matrices and Gaussian elimination. You’ve certainly worked with basis vectors — the i, j, k that many of you as engineers have undoubtedly worked with in your physics and dynamics courses. We’ll be covering those things here. But we’re going to look at them from a different perspective. That’s the advanced part of this course. We’re not concerned so much with how to manipulate matrices and vectors to solve systems of equations, rather we’re going to study the fundamental principles that underlie it all.

But this is not a math class.

Most of you are graduate students in the College of Engineering, and this course will give you as engineers the mathematical foundations necessary for your engineering coursework and research and careers. Although we will discuss proofs of theorems, and although I will expect you to demonstrate some mastery of a few of what I consider important and representative proofs, this class is not exclusively about proofs. That’s the for engineers part of this course. It’s about getting a mature grounding in concepts that maybe you thought you already understood.

To be honest, mathematicians will tell you that there is no way to master the fundamentals without mastering the proofs. And in some ways they are right. Still, this is an engineering course, and we obviously have a slightly different view.

We’ll start with concepts that are undoubtedly familiar: vectors and systems of linear equations and Gaussian elimination, but we’ll immediately step beyond those. We’ll study concepts like linear independence and spans and basis vectors, concepts which you might have encountered. But here we will dig much more deeply than I suspect you have before. And we’ll work with concepts that are probably new like homomorphisms and null spaces and duals and functionals. If you stick with me, you’ll get a much, much more solid understanding of not only what these things mean (I mean, what they really mean) but also how they relate.

I confess, this won’t be easy.

At every turn, you will be tempted to think that you understand after skimming over the material. And if you’re running short on time, that temptation will be easy to give in to. But don’t. Because to really master what we cover, you will need to do much more than skim.

Leave your undergrad days of cramming the night before behind you. You’re graduate students, now.

You need to immerse yourself in this. You need to let the concepts and their meanings (and yes, sometimes even proofs) wash over you until you begin to see everything as a single, unified whole rather than as a set of rules or procedures or algorithms. You need to bring a different attitude to mastering this material than you did to your undergraduate courses. This different attitude is what distinguishes graduate school from what came before. If you adopt this attitude, and if this deeper way becomes second nature to you, you will do well. And I will have done my job well.

So, welcome aboard!  Let’s dig in! …