Linear Algebra (LAFF) by UT Austin – Course Review

Course profile:

  • Link:                LAFF at edX
  • Professor:       Robert A. van de Geijn
  • Institution:     The University of Texas at Austin
  • Date:                Summer 2015
  • Platform:        edX

Linear Algebra was the first real mathematics course I decided to take for my current project. While the importance of linear algebra cannot be overstated, it is not exactly an accessible subject. I already took a Linear Algebra course in college, but I did not learn much (as usual, sadly). From my memory it was just another frustrating class that I stood in only to fulfill my degree requirements.

How It Went Wrong at College

My Linear Algebra professor at college focused the entire course on the techniques of linear algebra. These techniques included simple matrix-matrix multiplication, Gauss-Jordan elimination, projection, orthogonality, fundamental spaces, linear least-square, and eigen-blahblah. If you have some experience with the subject (and you really remember it), you should find those topics familiar. So in a way, my professor did teach everything he was supposed to teach. Except he did not.

The techniques, while indeed important to master, do not tell the whole story of Linear Algebra. It is not about applications either. I appreciated that my professor made some effort to give a recitation each week (most courses in my college did not even have recitation). He and the teaching assistants gave several presentations about the cool applications of linear algebra, such as image compression in JPEG file. Still this information did not help the students, including me, at all. The one thing that my professor forgot to teach, or deliberately avoided, is the same thing that should have made students fall in love with linear algebra: the insight.

Why LAFF Works for Me (and For You)

Techniques make up only a small part of LAFF (Linear Algebra: Foundations to Frontiers). This should not be surprising. With the ubiquity of matrix libraries as good as MATLAB, we can use a computer to apply the techniques. What we should do instead is teaching the computer to solve problems for us. In addition to computation techniques, this goal requires a deep understanding of linear algebra.

Prof. Robert van der Geijn, who teaches LAFF, insists that insight can only be gained by doing mathematics. In this class, that means understanding proofs and proving theorems. In fact, Robert make all the answer keys available at the beginning of the course. A student who just wants a certificate (like the old me in college) could just enter all the answers and get a good-looking piece of paper without even knowing what Linear Algebra is. The reason for this approach is it makes absolutely no sense to use the techniques without understanding the proofs. Proofs are not created for a specific kind of creatures called mathematician. They are important for anyone who wants to learn mathematics the right way.

Admittedly I did not dare to read even one proof in the linear algebra class in college. I even did not remember the professor mentioned any proof, or whether he included proofs in the homework and exams. (He did not!) Sadly, it was and remains the standard teaching method of mathematics nowadays. I will definitely write more about the topic of mathematics teaching in near future.

Back to the course, it was the first time I tried to make proofs by myself, without any external motives like homework or exam. It may sound extremely nerdy, but math has changed me, inside out. I’m a changed man. Suddenly I find myself interested in reading and understanding proofs. My attitude towards Linear Algebra and Math in general turns a complete 360 degrees. Now I am eager to learn more. It’s crazy, but finally it seems I have found my favorite subject.

Studying math should be like this. It should be about thinking, exploring, proving, arguing and imagining. It should be a creative process, not a mechanical one like what is taught in school.

I want to thank Prof. Robert van der Geijn, his wife, and his assistants for an excellent course. Really they taught me to love math.

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Book Review: Malcolm Gladwell – Outliers

I don’t think I have ever been working this hard in my life until the last few weeks. It is the first time I feel like working/studying from dawn to dusk (of course, that’s a metaphor since I can never get up at dawn). Today is Saturday, a perfect day to take a break and do something else other than write a mathematical proof. In this article, I will review a book by Malcolm Gladwell, Outliers: The Story of Success.

Below are my thoughts on several arguments made my Malcolm in the book.

The Matthew Effect or The Rich Gets Richer

When we look at a successful professional, be it a concert pianist or a renowned entrepreneur, we think he or she is special. We seem to understand how difficult it is to succeed in life. Everybody knows that we need both talent and a lot of hard work to achieve a great career. Nothing is wrong with that belief. It just does not tell the whole story.

In Outliers, Malcolm discuss a phenomenon of young ice hockey players in Canada. Somehow, the most promising players were mainly born in the first half of the year. It is absurd to believe the date of birth can affect your hockey skill. Indeed it does. Being born early in the year gives a kid a few more months of practice than another born late in the same year. A few more months may not sound a big deal for adult players, but for a five-year-old kid, that short period of time can make the difference. The better players are then selected into better teams with better coaches. These kids also have the opportunity to play with good teammates in competitive leagues. They accumulate experience fast and constantly improve as a result. Ten years later, we can guess who is better, the kid born early versus the kid born late.

Success requires good fortune, it seems. The question is: Can we create our own good fortune?

Work Hard and Watch Out for Opportunities

We cannot predict the future. However, we can prepare for it and wait for the right opportunity. How? The tentative answer is to choose a meaningful work and do it really well. But what is the definition of meaningful work? Let me try to explain it in a few sentences.

Meaningful work is something that not only compensates us well (not just money), but also contributes to society. Take Cristiano Ronaldo or David Beckham as examples, they do only one thing well – playing soccer – and that is enough to be a meaningful work. Their soccer skills entertain fans around the globe, and people readily pay big bucks to watch them playing. Meanwhile, no one ever will pay to watch me play, indicating professional soccer cannot be meaningful to me although I might be dying to play soccer day in, day out. In short, a meaningful work is a work that you enjoy doing, and it should serve people in some meaningful way, and people will reward you, usually in the form of salary and status. If you want to read more about meaningful work, Cal Newport surely has a much better explanation than I do.

In Outliers, Malcolm Gladwell lists several characters who have meaningful work: Bill Gates, The Beatles, Joe Flom among them. They shared a similar story. They worked really hard at first even though they never knew how successful they would be. They simply focused on honing their skills in their respective careers and look for opportunities to serve the society. Eventually they took the opportunities to shine and were recognized as successful.

Note that success is relative anyway. Success cannot be quantified. For me, success means that I have enough money to feed my family, and I enjoy my job. I like a simple and happy life.

Cultural Heritage and Its Influence

The last topic I want to mention is cultural heritage. Malcolm spends a significant part of the book to tell stories about different cultures. They vary from an Italian town in Pennsylvania where its residents are extremely healthy, another town in the American South where people are ridiculously violent, Robert Oppenheimer versus Chris Langan, to the alarming number of plane crashes of Korean Air. All of the stories convey the fact that many people are not aware of the power of their cultural heritage.

Cultural heritage defines who we are, influences what we do and how we think. I know this too well after coming to the US from Vietnam. The two cultures cannot be more different. I suffer from this difference even in simple, everyday conversations with Americans. I still don’t know how to handle a conversation in a large group. I used to blame my own personality for my failure to communicate. But truth be told, I just do not fully understand their culture as well as my own.

Adaptation is a crucial part of any success story. As an instance, the Jewish immigrants of New York in the early 20th century, as told by Malcolm, worked hard in negotiation skills. Despite the language barrier, many Jewish immigrants could expand their markets by negotiating with American capitalists. Right now, negotiation is only something I wish I could do.

This lesson, and not the popular “10,000 hours rule”, is what I appreciate most in Outliers. I will definitely try to work on my public speaking and negotiation skills. The time has come to grow up.

September Plan

This post is the first report of my current project. It will not exactly be a fun post but I want to write something anyway, simply because it’s better to jot down my thoughts once in a while. I plan to write more interesting articles in near future. I will write book reviews to summarize things I learned while reading. Maybe I will also write about the cool things about the courses I’m taking.

The beginning
Since initiated the project last month, I have finished two courses on Coursera, MATLAB Programming by Vanderbilt University and Algorithms, Part I by Princeton University and one mini-course on Codecademy, Introduction to Python.

Algorithms, Part I was a challenging yet enjoyable course. It taught me about the beauty as well as the importance of algorithms that I never realized before. I did take one data structures and algorithms course during college but I insisted on solving every assignment by brute force. I guess because my professor did not have a sophisticated grader like the one used by Dr. Sedgewick from Princeton, I managed to escape with perfect scores despite never applying the algorithms I was supposed to learn. At least it’s a different story now. Thanks to the machine grader on Coursera, I was forced to write clean, maintainable, efficient and sometimes even elegant Java code.

Still one thing Algorithms, Part 1 did not do is teaching the theory behind the algorithms since it only focuses on implementation. I have never been interested in theory, and math in particular. I used to avoid rigorous proofs like a plague. This is a good time to change. Fortunately, Dr. Sedgewick will offer the Analysis of Algorithms course next week. Let the math journey begin.

My goal is to be (surprise, surprise,) a math whiz

Speaking of math, I would like to discuss why I decide to change my attitude towards math. In the last few weeks, I have been thinking about all the gaps in my background, and why I have struggled immensely although I have spent a lot of time studying. It turns out my lack of mathematical maturity is the culprit.

I have two big issues relating to math. First, I do not have good strategies of attacking a problem methodically. In other words, I have always found it difficult to think clearly and make progress while working on a problem. I think this ability comes from working on the proofs in all the math courses: discrete math, calculus, linear algebra to name a few. It’s a shame that I never, ever cared about proofs.

Secondly, I simply do not have the math background to understand intermediate to advanced engineering materials. Stuffs like linear algebra, Fourier series, Laplace transform are the basic tools for any professional scientist or engineer. Because I never acknowledged the applications of these math subjects, I avoided learning them. My professors in college also did not help the cause by giving easy As. Now I have to re-learn all over again. Is it painful? Hell yes. But it must be a hell lot of fun too.

September Plan
Here is my plan for September. I will discuss some of them in more details in later posts.

Linear Algebra: Foundations to Frontiers, UT Austin.
Effective Thinking Through Mathematics, UT Austin
Algorithimic Thinking (Part 1) and Principles of Computing (Part 1), Rice University.
Game Theory, Stanford.
Analysis of Algorithms, Princeton.
Calculus I, Ohio State University

Brendon Burchard – The Motivation Manifesto.
Geoff Colvin – Humans are Underrrated.

Codename: Before Sunrise

It has been just over three months since the last time I wrote in my blog. Many things have happened during a great summer here in upstate New York. My girlfriend will start her PhD in two weeks. We are already settled in a nice, comfortable apartment in a small, quiet town.

Unfortunately, my career as a graduate student is totally in chaos. I really do not know whether I could or should continue with fundamental research.  Research is a too big ask for my current ability. Fundamental research requires a solid foundations in math, physics, statistics to name a few. Certainly I have not mastered the fundamentals well enough. I know I must rethink my career.

In the next 9 months, I will try to regroup by doing three things. First, I will study mathematics and physics from scratch, starting with Calculus, Classical Mechanics, Linear Algbera up to Electromagnetics, Integrated Circuits, and Stochastic Processes. Second, I will study Computer Science from ground up. I will learn C++, algorithms, data structures, web developement, etc. And lastly, I will try to read as many books as possible. After this period, I will try doing research once more and regardless of the result, move on with my life.

As usual, I will give this huge project a name. I will call it Before Sunrise, as “it is always darkest just before the dawn.”

Let me believe that it will be beautiful when the sun finally rises next year.

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The 2015 Challenge: Entry 10 – A Lesson of Persistence

This week has been tough. I spent a lot of time trying to make progress in my research, but ended up frustrated everyday. Even worse, because my research requires working with a computer, I found myself wasting time throughout the day by surfing the web randomly. Nothing is more dangerous to productivity than a habit of using Internet at work. Maybe the solution is telling myself to stop opening a browser during research, right? Unfortunately, the issue is more complicated; it demands a serious and not-so-simple answer.

Actually I begin each research session with a good mood. I usually have several ideas of possible solutions to the problem at hand. Then I implement these the ideas and pray they would work. But most of the time they fail, and I become disappointed. I repeat the cycle over and over for another 30 minutes or so, and if I do not get a satisfactory solution, I would be deeply frustrated. You may guess what happens next. A click at the Google Chrome’s or Firefox’s icon opens an escape for me, at least temporarily. This is not a great way to do research, I suppose. Something must be changed in my routine.

Yesterday when I collected my stuffs before heading home, I recalled a blog post by Matt Might, a computer science professor at the University of Utah. He has a few articles about PhD students that I really enjoy. In one of those articles, he wrote about the qualities every successful PhD student should have. Persistence is the most important one.

Let me quote a passage from the article:

To survive this period, you have to be willing to fail from the moment you wake to the moment your head hits the pillow. You must be willing to fail for days on end, for months on end and maybe even for years on end. The skill you accrete during this trauma is the ability to imagine plausible solutions, and to estimate the likelihood that an approach will work.

If you persevere to the end of this phase, your mind will intuit solutions to problems in ways that it didn’t and couldn’t before. You won’t know how your mind does this. (I don’t know how mine does it.) It just will. [emphasis mine]

Everything Matt said about persistence rings true. It is enlightening to realize that I have the right to fail when I do research. If I know that it is okay, even normal, to fail repeatedly, then I can feel better about myself. I will be more likely to persist working on the problem, generating other ideas, implementing them, and possibly facing a lot of failures. It sounds like a huge test of character, but PhD is exactly that.

Last night I also went to a talk by Steven Squyres and Bill Nye to celebrate the 150th birthday of Cornell. They discussed the joy (and the pain) of discovery. Steven likened his work for the Opportunity mission on Mars to an adventure. He and his team, all brilliant people, most of them having PhD degrees from prestigious institutions over the world, still had to endure countless failures during the mission. Many times they even had no idea of what they were doing. Still they persisted, and eventually they successfully landed the rover on Mars. That is a spectacular feat of discovery.

Research is discovery.  You do not know what will be coming your way. You just hope somehow you will figure all out. And you need persistence, a heck lot of it.

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The 2015 Challenge – Entry 9: Challenges Still Ahead

Last week I met Brian for a dinner. He was my partner in a class last semester. I went fishing with him several times (actually I watched him fishing and did little else). He told me that he’s ready to quit PhD for a full-time job. I was stunned for a moment. I knew he had been struggling for a while, but still it made me realize PhD is not for everyone. PhD is not college, not even close. It is tough to do your own works and overcome your own problems most of the time. Clearly Brian was not prepared for the change, even though he got a degree from the number 1 electrical engineering program in the country.

Fortunately, I will remain to continue, at least for another year thanks to the funding I recently received from my advisor. But that does not mean much in my 2015 Challenge. I created the Challenge not with the funding, but my self-improvement as the ultimate goal. Everything was going okay until I caught a cold during my visit to Dallas, and I was in a bad physical condition for the last two weeks. My research did not go as fast as I hoped, not only because of the cold, but also because the problem was getting harder.

Today I’d like to discuss about two aspects in my life that I’m working on. I believe if I make significant improvement in these two areas, I will finish the 2015 Challenge on a high note.

The first issue is sleeping. I have not had enough hours of sleep each day. After a few years I realize that I actually need a full 8 hours of sleep a night. However, my bad time management during the evening, my habits, and some personal issue collectively prevented me from getting to bed on time. I am still too attached to the computer, and the time I spend surfing web has occupied nearly the whole evening. I have done the first step as I have opted out of Facebook. I also force myself to read and practice playing an instrument, 30 minutes of each activity per night. Of course, getting rid of a nearly 10-year-old habit will be challenging. At least I have experienced the amount of energy I should have after an eight-hour sleep last week. That will be the motivation for me in the upcoming month.

The second issue is related to the first: I need to have the energy to battle the problems in both research and books. It demoralizes me each time I read a problem and have no idea how to solve it, although I think I understand the book chapter well enough. I often worry that I will never have enough time to conquer the problems and to gain the insight of the topic in question. And that means I will fail to catch up all the smart students in my research group. Nevertheless, I know I have to keep working and make progress, step by step. I know too well that if I avoid suffering now, undoubtedly I will regret later. I determine not to repeat my mistakes in the past. Everyday I will remind myself that I need to suffer now and reap the rewards later.

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Money Can Buy Happiness

Last week I came to Dallas to visit my alma mater, my girlfriend, and a host of friends. It’s always good to back at Dallas, where I consider as my second hometown after Hanoi. During my time there I was reminded of many memories at UT Dallas. One of them is the remark that I was notorious for spending too much money, and somehow I refused to apply for any on-campus job during college. Even worse, my family might actually be one of the poorest among my countrymen at UT Dallas. How ironic it is!

In a sense, my friends are right. I did buy a lot of stuffs: headphones, speakers, tennis rackets, table tennis paddles, soccer cleats, a car, and numerous posters. Funny enough, most of those stuffs do not aid in my survival at all. Surely I don’t need different headphones to pass exams. Many posters also have nothing to do with my job prospect. Why I seemingly wasted money on them?

Let me give you an example: car.

Car Brings Me Happiness

The common sentiment among my friends at UT Dallas is you do not need a car unless you have a job. It sounds reasonable at first. A car costs quite a lot of money. You need to set aside at least around $1500 a year for insurance, gas, and maintenance, let alone the original cost of the car. It is understandable that many of my friends cannot afford a car. It only perplexes me when several guys and girls, who come from rich families, do not bother buying a car at all (or until their senior year). This line of thought leads them to think that for a guy like me, who has no immediate need for a car, owning one is wasteful.

So what is my reason of buying a car? Simple, I need to occasionally escape the campus and entertain myself. From what I experienced, nothing is better than a trip to Tapioca House on the weekend for some boba tea or to White Rock Lake for sight-seeing. A car is handy when I need to find some good food outside the campus, when I want to visit an art gallery in downtown Dallas, or when you drive a car just for the sake of it. After two years of having a car, I enjoyed great moments, beautiful scenes and a sense of freedom. My car really brought me happiness, a lot of it.

It is a different story when you only buy a car when you are a senior. Several of my friends, after years without a car, are still content to spend every night at the computers. They do not like driving that much. Maybe they are so busy with senior-level classes and job searching and do not have spare time for exploring the city. Maybe their habits have already formed a tendency of staying home every day. I don’t know, and I don’t care. All I know is I’m happy with my decision to buy a car early.

Use Money Wisely

I believe that spending money wisely is not easy. Many people, however, focus solely on making money. But that does not mean you should avoid spending money at all cost. If your family struggles to support you at college, then I can understand. On the other hand, if you are well-supported, you should use money to your advantage. Use money to enrich your life. Use money to buy yourself happiness.

Some of my friends may wonder why I always find it easy to laugh, have fun and enjoy life. This blog entry might be one answer to that question.