Answer by Senia Sheydvasser:

There are some good answers on this question already, so I think I will answer this by giving a cute application of linear algebra over the finite field [math] \mathbb{F}_3 [/math]. If you aren't familiar with this field, it consists of three elements: [math] \{0, 1, 2\} [/math], with the convention that if you add, subtract, or multiply, you reduce the answer mod 3. So, for example [math] 2 * 2 = 4 = 1 [/math], [math] 1 + 2 = 3 = 0 [/math]. Linear algebra over this field works more or less exactly like it does for linear algebra over [math] \mathbb{R} [/math] and [math] \mathbb{C} [/math].

Now, let me go on what may seem like a non sequitur, and talk about the card game Set (Set (game)). Each card in Set has an illustration of one, two, or three or three symbols (which come in three different varieties). These symbols can be green, blue, or purple. Finally, they can be colored as an outline, shaded, or solid. Here's a picture:

The game is played by laying down 12 cards, and searching for a collection of three cards called a "set." Three cards are a "set" if, for each of their attributes (number, symbol, color, shading style), all three cards are the same, or all of them are different.

Thus, for example, the picture shown above would be a set (the numbers are all different, the symbols are all different, the colors are all different, and the shading styles are all different). But we could also, for example, have the same example, but make it so that there are three symbols on each card—this would still work.

Whoever collects the most sets wins. Simple, and on the face of it doesn't appear especially mathematical. But that is not so!

You can associate each card to a vector in the four-dimensional vector space over [math] \mathbb{F}_3 [/math]. The number of symbols will be one coordinate, the symbols themselves (say, 0 = rhombus, 1 = squiggly thing, 2 = rectangular ovoid), the color another, and the shading type another.

Then it is very easy to check that a set is just a collection of three vectors that sum up to the zero vector! Now

thatis a beautiful description, and suddenly it means that we can use the tools of linear algebra to answer questions about the game Set. For example, how many cards do you have to go through to ensure that you get a set? (21 cards.) And, thinking about it this way makes it a little easier to get really,absurdlygood at the game. I never played enough of it to get decent, but I have known some absolutely monstrous players (all of them math PhDs now).If you want to read a bit more about the mathematical background of Set, I suggest you look here: Page on rutgers.edu. (It seems the original link is dead. JSTOR has this article, if I'm not mistaken, but unfortunately you need to pay to download it.)

What is the joy of learning linear algebra?