In the previous post, we introduced the Cauchy-Schwarz inequality purely as a statement about real numbers:
We proved this inequality using mathematical induction. However, one could argue that the method did not yield much insight. As our proof was entirely algebraic, it is not easy to see why the sums that appear in the inequality are reasonable quantities to consider, and there is no hint as to how they could arise naturally in mathematical problems. Therefore, in this post, we will look at the Cauchy-Schwarz inequality from a geometric perspective, which reveals a much richer structure. Moreover, we will see how the inequality arises naturally from a basic optimisation problem.
Note: I continue with the convention from the previous post that a quantity is called positive if . We say that is strictly positive if .
After a long time, I have decided to resurrect the mathematics part of this blog! I started Diversions in Mathematics as a way for me to try to explain mathematics to the general public. This continues to be the main goal of this series of blogposts — for a more detailed introduction, please read the introductory remarks. I wrote two posts in this series, but then abruptly stopped. Part of the reason was that my studies got in the way, but I was also unsure exactly what material to present, and how to present it. I wrote down some of my thoughts on this matter in a previous update post.
One of the main issues as a writer is to consider the readers’ background in mathematics. For posts targeting the general public (like the previous two Diversions), I have tried to assume as little as possible while maintaining the discussion at an intelligent level, i.e. without “dumbing down” anything) However, this already assumes familiarity with many mathematical concepts taught in high school, or at least, some level of maturity in regard to abstract reasoning. Consequently I have decided to relaunch Diversions in Mathematics with high school mathematics as a foundation.
In this blogpost, I will introduce the Cauchy-Schwarz inequality, one of the most fundamental results in mathematical analysis, with the aim of connecting various topics that are typically studied in the Year 12 HSC maths curriculum in NSW. The main article is below.
Let me tell you about a mistake I made recently. Some months ago, I made a few calculations which seemed promising, and I thought they could lead to some new results. When I showed the calculations to my supervisor, he was skeptical, and rightly so. Our project involved some fairly abstract concepts in the theory of operator semigroups, so his immediate advice was to check the calculations against some simple, concrete examples . So of course I did that, and quickly realised that the calculations were totally wrong! Moreover, they were wrong for a rather silly, elementary reason — a pitfall that I thought I should be capable of detecting with my current mathematical experience.
To clarify what I mean by “elementary”: no, it wasn’t a silly error in the sense of a missing minus sign or multiplying two numbers incorrectly. Mathematicians say that something in their field of study is “elementary” if it is a basic fact in that particular field. However, to paraphrase my supervisor, “basic does not mean easy”, so if I were to describe the mistake to a non-mathematical audience, it would still be extremely difficult. Another interesting point is the following. When you do research in pure mathematics, quite often you deal with abstractions upon abstractions, and it is easy to forget the humble origins of these abstract concepts. For example, the concept of a linear operator is fundamental to many areas of maths, and as cool as it is to be working with linear operators on infinite dimensional Banach spaces or whatever, it is also worth checking a statement about abstract linear operators on, let’s say, a 2-by-2 matrix as encountered in 1st year university maths (or even high school, depending on where you were educated). This comic sums up the sentiment quite well (excuse the pun).
Now this brings us to the main point of this blogpost. The wrong calculation I described above is not the mistake I want to write about; rather, I want to focus on what happened next. I assume that most people do not enjoy looking at their failures. I am not sure what I did with those few pages of calculations: either I threw them out, or I shelved them away somewhere I had forgotten. In any case, recently I wanted to return to the same problem with a different perspective, but now I have lost my point of reference. Because I had misplaced or thrown out the incorrect calculations, I cannot really remember what I had got wrong the first time! To be fair to myself, I have a vague idea of the kind of error it was, and it is very unlikely I will make exactly the same error again, but nevertheless it would have been more efficient to keep the original calculations, wrong as they are, so that I can remember exactly what the lapse of judgment was. As I have learned by now, mathematical work is not only about putting one true statement after another. It is equally important and even illuminating to know what kind of result cannot be true. In hindsight, throwing away those calculations was a bigger mistake than the calculations themselves.
There are certain similarities with the performing arts, although I have to admit that the comparison gets fuzzy on closer inspection. For a start, mistakes in mathematics are objective errors: you are objectively wrong if you claim that 1 + 1 = 3, for instance. On the other hand, speaking from a background in Western classical music, even though there are certain objective errors (e.g. playing a wrong note), usually we are unhappy with a performance because we feel that we could have done certain things better. One could say this is a “subjective” error. Despite this fundamental difference though, both mathematicians and performing artists undeniably improve by learning from their mistakes, objective or subjective. Indeed, I would say that learning from mistakes is an essential part of development in the two disciplines.
As a violin student, I used to dislike making recordings — what a way to highlight one’s faults and weaknesses! However, in hindsight I should have taken more opportunities to record my playing, and to embrace whatever mistakes (objective or subjective) occurred during a performance. One should not take this viewpoint too pessimistically. The idea is not to focus only on failures; rather it is about using those failures strategically to improve one’s understanding or problem solving technique in the case of mathematics, or one’s expression capabilities in the case of the performing arts.
I end this post with a nice little comic, which comes with some wise words from the great mathematician Paul Halmos:
It seems that every time I return to write something on this blog, it’s about how infrequently I write (and that I’m awfully sorry, and I promise to write more, but it never happens). I see three main reasons for this:
I am now in a masters program in the School of Mathematics at the University of Sydney! My honours year went quite well overall, and my supervisor was happy to take me as a postgraduate research student.
Since March, I have also been working casually for Matrix Education, a private tutoring company that is well-known in particular for its HSC preparation courses.
From the two reasons listed above, you can appreciate that most of my time is spent doing mathematics in some way — reading articles and learning new theory for my research project, working on exercises from textbooks, or preparing tutorials and classes. The third reason is really a blend of two:
My own laziness coupled with a lack of direction about this blog.
This update post will explain this last point further.
On 19 and 20 January, I had the pleasure of teaching at the Zhang Violin Summer School for young violinists (AMEB grade 3 to A.Mus equivalent). This was a 4-day intensive course, where students received training in preparing solo repertoire and chamber music, as well as Dalcroze eurhythmics. I was invited to take chamber music lessons, a task that I gladly accepted. There was quite a wide range of ages: the youngest students were in years 1 and 2, while the eldest ones were senior school students. As a result, there was a similar range of experience in chamber music. Teaching at the summer school has reinforced my belief of the high importance of chamber music in music education, and in this blogpost I would like to share some of my thoughts on the subject.
First of all, I remark that it is a happy coincidence that the summer school was held at Barker College, where I was a former student. It was at Barker that I first experienced playing chamber music — specifically, the string quartet. As it turned out, I played string quartets throughout my entire education at Barker College, and it was one of the most important factors in my development as a musician — not only as a performer, but also as a composer! The main point that I will emphasise is this: playing chamber music makes great demands on the musicians involved, in technical aspects (i.e. the physical part of instrumental playing) as well as conceptual (i.e. intellectual understanding of the music). These demands are quite different from those encountered in solo playing, and furthermore, I will try to argue that chamber music should play a significant role in a student’s musical development.