Gram-Schmidt Theorem (idea)

Still, I prefer to read stories. I'll tell you my story on this theorem: how to spoil geometry.

You see, in primary school I thought learning was all about concepts and principles. You can say I had a Platonic mind. I knew how to add and subtract before I went to school; it may have taken me a short while to understand the principle of multiplication and that of division; I completely missed the point of having to practise these things for years on end. The ideas of mistakes and of acquiring skills were unknown to me. As soon as I thought I understood something, I would simply stop and refuse to do any more exercise on the subject.

So mathematics was incredibly boring. I remember asking my uncle whether there was a 'next step' that is to multiplication and division as they are to addition and subtraction, and he explained exponentiation and square roots to me. Since I remember the occasion, I can put a date to it and I realise I was 8 at the time. It was years before we ever heard of these things in school. Looking back, I was fascinated by mathematical laws, but bored stiff by calculations.

(Sorry, this self-rant was reserved for the unfinished school algebra writeup and some of it should move to my home node. I'll reorganise it when I have more time.)

So imagine my delight when in sixth grade - I was 11 - the world of Euclidean geometry was unfolded to us. Looking back, the exercises in geometry - proving theorems about triangles, lines, circles, etcetera - are easily my favourite subject among everything we ever learnt in school, including high school. It's not that I was particularly good at it, but abstract mathematics was my favourite subject. Why oh why does 99% of all mathematics have to consist of the hard and boring manual labour, the calculations! I hate calculations! I can't be bothered, and even when I try, I'm too sloppy to ever do them right.

In high school, the pattern continued: in mathematics and physics, some fundamental laws and their relationships would be introduced, after which you would get weeks, months or even years of exercises doing mindless calculations with those rules. Where was the insight? There was no place for that. After a while I became convinced that the calculations were the essence of physics, that my idea of 'insight' was a stupid romantic craving I had to shake off. Thanks to our teachers and books, I didn't leave school with the same feeling about mathematics: they never made me forget about the incredible world of pure and undiluted ideas that is in there, and I realized there would be more space for them in university, but, pathetic as I was at calculations, I didn't have the guts to study mathematics.

But in the first two years of high school, algebra and geometry almost had equal time. The algebra part - calculations calculations calculations - was just boring routine, but there was the wonderful part, the part that everybody else hated, the geometry - proving theorems, discovering things, using your imagination! (I think they no longer teach geometry in high school. Abstract thought is no longer considered a tenable goal of teaching these days.)

So what happened to geometry? They turned it into algebra! It's straightforward: you map out your geometrical objects in n-dimensional space, write down all the numbers you can think of, calculate, and the answer will appear. There's no need for your fancy theorem proving, just do the math, stupid!

Now read the theorem above, because this is what it says, albeit in formal mathematical terms: if a problem is subject to the laws of Euclidean geometry, it can be mapped out into an n-dimensional Cartesian coordinate system where the laws of linear algebra apply, and you can just 'do the math'.

It's not fair! Where did the abstract thinking go! Why does everything have to be boring calculations in the end! Mommy, they are cheating on me!

I'm exaggerating, of course: the abstract mathematics of algebra is just as rich and fascinating as geometry, intelligence and insight can be of use in doing proper calculations, and without any level of proficiency, without mastery of basic skills, you probably won't make much progress in understanding abstract theory. So there really isn't a full opposition between 'pure ideas' and 'do the math' in the way that I felt at the time. And I was aware of that, but it left me feeling sour with the exact subjects, and disappointed with my own abilities, and I never really recovered from that.

Don't get me wrong, I'm not blaming Gram or Schmidt for my problems with mathematics in highschool. You can't blame truth! Actually, we're dealing with a beautiful and useful principle. But it is a principle that more than anything else represents the loss of my faith in my own mathematical abilities.

If you're still here, I've been trying to convince you that mathematics is more than formulas: these formulas actually mean something - not just in an objective sense, but even in a personal sense. This is true for brillant and famous mathematicians - read Simon Singh's account of the story of Andrew Wiles and Fermat's Last Theorem - but also for the pathetic whiners who never even tried - as this writeup tries to explain.