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Advanced Linear Algebra for Engineers

Sunday, 14 Sep 2014, 12:22 UTC

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

Saturday, 13 Sep 2014, 10:40 UTC

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.

The Girl in Blue

Saturday, 13 Sep 2014, 09:06 UTC

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.

That Last Light of Day

Tuesday, 09 Sep 2014, 19:35 UTC

It was hot this afternoon when the sun was up. And running the hills was grueling. My pace couldn’t have been more than a walk on that last climb.

But now it was over. The sun had just set behind the hills. A breeze was blowing in the open windows of my car as I drove home.

The sky was clear. The evening air was cool … ish. And the skyline of downtown caught the last rays of the sun. Er, no. Did I say caught? There was no “catching” involved. It was far more than that.

The last light of day reflected off the towers. A pink-orange-white bright light glistened and glimmered and grew to an intensity so bright that I couldn’t stare. It was like… It was like… like the (wait for it) flame of a rocket.

And then… at that very moment, KC and the Sunshine Band was playing on the radio. I reached down and turned the volume way up.

The wind was blowing against me. The last light of day shined brightly. I bounced in my seat and sang with the song. And goosebumps ran down my arms.

And on that note… Let’s start this conversation back up, shall we? 


Tremendous Man

Monday, 11 Aug 2014, 18:03 UTC

It was raining on the drive today. Gray skies. Wet roads. Slow traffic in both directions.

There aren’t many things that can make me audibly gasp when I’m sitting alone in the car. The news of his death under wet, grey skies did. What a tremendous loss. What a tremendous man.

The Moon, Saturn, Mars and Spika

Monday, 04 Aug 2014, 20:46 UTC

Taking the garbage out to the can, I looked up. The sun has just gone down. The western horizon was still bathed in daylight. The sky overhead was a darker blue. I walked out into the middle of the street. From there I could see the crescent moon.

The moon, making its backwards way along the ecliptic has just passed Spika and Mars and Saturn, and I wanted to know if they were visible, yet. They weren’t, but I looked a bit longer, because in an early evening sky, the first stars have a way of hiding from you. So I stood there in the middle of the street (having looked both ways as our parents taught us). I saw nothing, so I went back inside.

A few minutes later, I came out again — another handful of garbage for the can. 

The light of the sun over the horizon was fading. The eastern sky was that deep indigo blue. (I know you know the color.) The sky overhead was darker than it had been the last time. So I wandered out again into the street. And there, just to the west of the moon was a dim, barely visible dot.

Would that be Spika? No, too close to the moon, which was just west of Spika several days ago. Not Mars, either — not red enough. It must be Saturn. Then where’s Mars? I gazed further to the west and saw nothing. Oh for heaven’s sake, where’s Mars? I looked harder, because … well you know … they hide from you. And sure enough, there was red Mars. Ok, then, where’s Spika, then? Bright Spika, after all, sits stationary in the ecliptic, masquerading as a planet, fooling the unwary. And sure enough, there was Spika.

Bang, bang, bang, bang. The moon, Saturn, Mars and Spika. One after another. Four in a row following the great arc of the ecliptic spread out overhead. You could almost see the curve drawn across the heavens.

I dropped my second handful of garbage into the can, and went back inside.

Rolling on the Cobblestones

Sunday, 15 Jun 2014, 20:01 UTC

1. After the Race

The sun was going down. The Canadian springtime evening was getting cold. We changed out of our sweaty race clothes, walked towards Rideau Street and watched the sun set behind the hills across the river in the west.

Then we wandered to Byward Market for something to eat, settling finally on The Highlander Pub. Best fish-and-chips I’ve ever had.

We sat on the patio. We were in no hurry. We sat and watched the world go by as we waited for our food. And when our batteries were finally recharged, we paid our bill and walked down George Street toward the bus stop on Dalhousie.

2. Rolling My Ankle

It was quiet there compared to the patio outside the pub. And it was dark. We stepped between two parked cars to cross the cobblestone street.

And here is where it happened.

I stepped off the curb, a step that was only an inch or two down, and I rolled my left ankle. My leg folded out from under me. Hands in pockets, unable to catch my balance, I fell into the street. Hard, elbow slamming directly into the cobblestones.

What I remember is this: a brief instant of confusion, the impact of my elbow, the clang of my race medal bouncing off the street, and my head bouncing twice off the ground. Thankfully by the time my head hit the ground, my elbow has broken my fall, so I was fine. But I had turned my ankle hard, and my elbow hurt. 

3. On Not Standing Up

“Are you ok!?” Trudy asked, relieved when she realized that the street was barricaded and so there was no rush to pull me out of oncoming traffic.

“Yes,” I said, struggling to sit up. Then the world went fuzzy.

The blood raced out of my head down to my ankle. You might know this feeling: stars at the periphery of your vision, tunnel vision, ringing in your ears, the smell of losing consciousness.

I knew that the last thing I needed to do was to stand up.

I scooted myself inch by inch back onto the curb up against a tree. Or was it a parking meter? Anyway, I sat there for a minute. Several minutes. No, many minutes. No, enough minutes that we didn’t catch the bus and ended up having to wait 30 minutes for the next one to come.

4. Evidence to the Contrary

Why am I telling this story?

About a week ago I sent a friend a note telling her about rolling my ankle. She’s a runner (a serious one), and I thought it would be a good runner’s commiseration. And since the Trudy features prominently in the story (having been prepared, after all, to pull her fallen husband out of the street), I CC’d her, too.

In this message, I said, and I quote, “Even though Trudy was the only one who had a beer, I fell into the street…”

The next day, Trudy walked into the study in the evening and looked at me.

“Do you really not remember having a beer that night?”

I stared blankly back at her. I searched the vaults of my impeccable memory. I drew a blank.

“No,” I said, “I don’t.”

“You had a beer,” she said. At which point the dimmest of memories returned, a memory of tasting her beer and ordering one myself. And although I honestly don’t remember any more than that, I suspect that if those fish-and-chips were so incredibly good, I must have really enjoyed that beer, too.

The day after that, I got an email from the fair and industrious Trudy. There was a photo attached. I submit that photo here as documented evidence. There you’ll see my 10K race medal. And there you’ll see me gripping the silverware in anticipation of those fish-and-chips. And there you’ll see the reflectors on my running jacket shining in the flash from Trudy’s camera. And finally, yes, there you’ll see an almost empty glass of beer sitting undeniably close to my right hand.


So there is no question. I did drink a beer. 

Now of course, that beer had nothing to do with rolling my ankle on George street, but at this point I think my credibility is shot, so I’ll just sit down.

Gatineau Park

Saturday, 14 Jun 2014, 22:04 UTC

Just outside Ottawa across the river in Quebec there’s a wilderness. Turn left after you cross the bridge, drive up the sloping road, and turn right. The city disappears behind you. The forest envelopes you.

We went there to hike two trails that the fair and industrious Trudy had selected from the many in the park.

We went to Champlain Lookout and gazed out over the farms and woods and river bottom land that once upon a time a very long time ago was Atlantic ocean washing up against the very bluffs we stood on. We sat and enjoyed the breeze and ate snacks and then turned to follow trail number 9.

Now if you look at a map of the Gatineau trails PDF, you will see that the trails wind around and thru creeks and ponds and lakes. At each turn you are met with rivulets or still pools.

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And at each turn we were met by swarming clouds of mosquitos and black flies. “It’s that time of year,” they told us later at the visitors center. “Silly Texans,” we thought to ourselves. No wonder we had the trail to ourselves.

So there we were in this wilderness of green leaves and dark waters and … is that a what I think it is across that pond? Anyway, there we were in the midst of this stunningly beautiful place on a stunningly wonderful spring day, and because of the mosquitos and flies assaulting us, we could barely stop to take pictures of each other or the wonders we found at each turn. And yet…

We descended into low, swampy areas where the Garter snakes slithered out from under our shoes.

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And we saw dark groves of Hemlock trees soaking up the light of day.

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And Ferns uncurling.

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And flowers… If only those mosquitos… but the flowers, just look at them.

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But oh those darned mosquitos!

The Kindly Hosts that Brung Us

Saturday, 14 Jun 2014, 13:15 UTC

Kindly hosts

A Walk in the Park

Thursday, 12 Jun 2014, 20:58 UTC

We left to go for a walk in the park, dad, Khadija, Trudy and I. But the roads and bridges were closed for the full and half marathons. Yet, they have plenty of parks in Ottawa, so there were many possibilities, and we gradually made our way to Andrew Haydon Park on the Ottawa River.

There were sailboats moored in the harbor at Nepean Sailing Club on Lac Deschênes.

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There were lagoons surrounded by deep green lawns and trees and pink blooming Crabapples

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which Crabapples we don’t get to enjoy very often, so we did.

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There were Canada geese, parents and goslings in a row.

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And finally, oh look. There across the lagoon: the kindly hosts that brung us!

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