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Ok, so let me give you the background story.

In my country, unlike the US, there are different high schools for different levels of 'intelligence'. I believe there are 5 or 6 types, the toughest being 'gymnasium' which is for the top 5% I believe.

I am currently enrolled at a 'gymnasium', and they have a special program for the most gifted out of the gifted. I am in this program, and in this program we basically get a couple of hours a week off to work on a presentation for the entire school (in April every year).

Most of the children participating in this program just do nothing in the hours they take off and just give a musical performance, but I wanted to do it about Linear Algebra this year, since I've started a self-study almost a week ago (now I'm learning about null spaces). (Sub-question by the way: Am I fast or not? I started L.A. 5 days ago, never having seen a matrix in my life and now I am learning about null spaces.. but it seems I am going too slow (or maybe that's just because I'm so excited and want to learn more and more every day)).

My main question is:

What would be interesting in a presentation of Linear Algebra? What concepts, ideas, etc. are mind-blowing and would be nice (for an intellectual crowd, don't worry about people being completely disinterested in math).

You can give an advanced answer, it's October and I still have half a year!

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There are some interesting applications of linear algebra. I don't think its mind-blowing, but one option could be linear optimization. For example, check this out: en.wikipedia.org/wiki/Simplex_algorithm –  Johnny Westerling Oct 9 '12 at 17:13
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I would begin with learning how to use line breaks. –  tomasz Oct 9 '12 at 17:41
    
A historical development of linear algebra would also make an interesting presentation. Ideas of linear algebra such as systems of linear equations, determinants and vectors have been studied for centuries, but it wasn't until quite recently that the ideas started to converge into a single central field. You might want to talk about the unifying concepts behind linear algebra and what lead to their developments. –  EuYu Oct 9 '12 at 18:22
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Tomasz advice is really good. Any time you present yourself in writing you certainly have to do better than a single giant jumble of sentences. Better writing organization should aid the quality of your speaking presentation as well. No matter how good the contents of your presentation are, it's going to be bad if it's poorly organized and presented. –  rschwieb Oct 9 '12 at 18:26
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tomasz was a bit blunt, but what tomasz and rschwieb say is true. No matter where it is presented, one big block of text is less likely to get attention than a nicely formatted and organized article. You may not see it, but they are trying to help. –  robjohn Oct 10 '12 at 4:22
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Probably not the most amazing, but fairly easy to explain to a non-nerd crowd: finding area and volume using determinants.

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I second the idea! And may I suggest presenting a geometric (informal) proof of Cramer's rule? –  Tunococ Oct 10 '12 at 1:16
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There is a lovely little book by Jiri Matousek, called Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra, published by the American Mathematical Society.

EDIT: Here are some of the topics discussed in this book:

If every club in Oddtown must have an odd number of members, and every pair of (distinct) clubs must have an even number of members in common, then the number of clubs can't exceed the number of citizens.

Error-correcting codes.

What's the largest number of lines in 3-space such that the angle between every two of them is the same?

A rectangle $1\times x$, where $x$ is irrational, can't be tiled by finitely many squares.

The Matrix-Tree Theorem.

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