…and now I’ve got to go to bed. So sayeth the Fair and Industrious Trudy.
…and now I’ve got to go to bed. So sayeth the Fair and Industrious Trudy.
You saw him, didn’t you? Governor Poindexter up there on the stage looking serious, speaking to the cameras, taking charge of the epidemic that’s gripped the nation, posing in front of that Texas hospital that was in the vanguard.
Except that the hospital didn’t manage things quite like they would have liked. Not quite as the governor would have liked, perhaps. And stop for a moment. Think about it. When is the last time you saw Governor Poindexter saying serious things and gazing thru his serious glasses from the hospital podium with the staff and administrators gathered around? When was the last time?
Funny that. He seems to be taking a trip to Europe now. Being in the vanguard seems to have seriously lost its luster.
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.
Julia didn’t sleep at all that night. Even though Jack and Katherine fell fast asleep. Even though they had been up late, and she should have been tired; she was not. And she didn’t sleep a wink.
The images and the sounds of … that place … swirled through her head. She could smell the sea breeze. She could feel it blowing thru her hair. She could see the stars against a black sky. All night long.
The next morning, she sat bleary-eyed over her cereal while Jack and Katherine devoured theirs. She just stared at her bowl and swirled her spoon.
“Good Job, Jack!” her mom announced as her brother got up from the table. Jack was skinny and always burning more calories than he took in, so they were always trying to get him to eat.
“You, too, Katherine!” Julia’s sister was finished and had gone to investigate something she saw moving in the underbrush at the edge of the woods.
The weather was perfect that day. The sun was shining against a blue sky with white clouds blowing on a breeze out of the west. The sunlight glistened on the water. The air was warm, and they went swimming.
Afterwards, they sat on the dock. All of them except Jack who was still running into the water and out, burning his calories. Katherine was bundled in a big towel shivering in the breeze. Julia was dangling her legs in the water, watching minnows dart in the shallows. Ben sat next to her. Ben was their cousin, Jack’s and Katherine’s and Julia’s.
“Whaaat’s wrrrong?” he asked, drawing out his vowels, splashing water on Julia’s legs, bumping her with his shoulder.
He had been on the couch the night before when they were all telling stories, when she told her story. When she saw … that place. Julia looked up and mumbled something.
“No, really,” he said. “What’s wrong?”
“You don’t want to know,” she said.
“Oh, but I do,” Ben said with a drawn-out “dooo” and a big smile on his face as he leaned into her again.
She didn’t want to say anything. She didn’t want to tell. But in the end she did.
She told them about what she had seen. About the big hole that had opened up in the air right in front of her that night. About the waves crashing in the distance. About the stars in the night sky. About the sea breeze. And about how she had to run to get back out, to get back into the room in the cottage where they had been sitting.
“What!?” said Jack. He had ceased his charging up and down the beach and had come over to listen to Julia’s story. “What!?” he said with his eyes wide open and his jaw agape.
Katherine rolled her eyes. She had this knack of silently rolling her eyes to show indifference or disbelief or disinterest. She clearly didn’t believe a word Julia had said.
“You don’t believe me,” Julia said. “I shouldn’t have said anything.”
Katherine rolled her eyes again. Julia got up to leave.
“Wait,” said Ben. “Don’t go. I believe you.”
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…
What a cranky snob I am.
They’re staring at me. Both of them. Just sitting and standing there staring silently directly at me.
It’s only 4:30. No, you can’t have dinner yet.
That’s what I get for feeding them early before I went out last night.
And darn it, now I’m hungry, too! Thanks a lot, doggies.
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! …
In the drizzle out there somewhere is a beetle trying to stay dry. That beetle that was lumbering along on that log beneath the Live Oak trees. Lumbering along and then climbing up a Morning Glory vine in search of … something.
And in the drizzle out there somewhere is a lizard trying to stay dry. That green lizard that turned to brown as he staked a claim on the fallen leaves beside the ant hill and grabbed the flying queens as the emerged. Grabbing them one after another and making a feast out of it.
In the drizzle trying to stay dry they are, I suppose. Or not, because it’s not every day that we get a drizzly day like this, and maybe they’re reveling in it as much as we are.
If it would only go on for a week.
I thought we were starting the conversation back up.
So did I.
Well, where have you been. What are you waiting for?
How about you tell us a story.
A story. Well, it just happens I have one to tell.
Perfect. I’ll sit here and be quiet. The floor is yours.
This is the story about a girl in blue…
It was dark outside with only a dim hint of day left beyond the trees in the west. The kids were sitting on chairs, on the sofa, in a circle of sorts. One of the boys had been reading a book. Now, a girl in blue sat up.
“Once upon a time,” she started.
The others turned their heads. They were looking forward to a story, but she was going to play a trick.
“Once upon a time,” she said with a smile on her face. And now that she had them, she was going to abruptly end with a quick “The End.” But before her words got out, she looked up and stopped short.
The air shimmered and shivered, and just beyond her reach a dancing pattern of cross-hatched lights appeared, seemingly tearing the air apart, opening as a zipper might unzip. She stared, dumbfounded, as the air became dark. And she heard the sound of crashing waves. The unzipping darkness grew.
She looked around the room. No one was moving. Not as if they were staring at the blackness but as if time was standing still.
She stood up and took a step toward the blackness. There was a humming sound as she got closer, and the blackness spread further. It was now as big as a door, and she could see into it. There were stars against a black night sky. There was grass lit by moonlight. There was a sea breeze blowing in her face. And from somewhere in that night she could hear waves breaking against a shore.
She looked around her again. No one in the room had moved.
She took another step toward that unzipping night. It opened further and in a moment she found herself on the grass in the moonlight under the stars with a breeze swirling around her face blowing her curls into her eyes.
The turned around. There was her chair. There was her sister sitting in the couch. There was the kitchen with someone standing at the stove. But no one was moving, and… they were getting farther away.
With a panic, she ran back. Back to her sister. Back to her chair. Back to the kitchen. It was already many steps away from where she had been standing. She ran as fast as she could.
Then in an instant she found herself not just back in the room, but back in her chair sitting where she had been sitting.
“Once upon a time,” the words has just come out of her mouth.
She jumped up. There was no blackness in front of her. No moonlit grass. No stars. No sea breeze or crashing waves.
Her sister and brother and cousin were there in front of her with surprised looks on their faces, because she had jumped out of her seat. Her sister was fidgeting in the corner of the couch. Her brother’s eyes were wide and his mouth was open in anticipation. The sound of popcorn popping was coming from the kitchen.
“Once upon a time,” she said, “the end.” And she sat back down.
There were groans from the crowd, who were expecting a real story. There was a little bit of a smile on her face for the trick she played.
Or rather she had thought it was going to be a trick. She had thought it was just going to be a quick story. But the story was in fact just beginning.
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