The Set Tour, Part 3: R^n
After talking about the real numbers last time, I had two obvious sets to use as follow up. Of course I’d overthink the choice of which to make my next common domain-and-range set.
Rn is pronounced “are enn”, just as you might do if you didn’t know enough mathematics to think the superscript meant something important. It does mean something important; it’s just that there’s not a graceful way to say what offhand. This is the set of n-tuples of real numbers. That is, anything you pick out of Rn is an ordered set of things all of which are themselves real numbers. The “n” here is the name for some whole number whose value isn’t going to change during the length of this problem.
So when we speak of Rn we are really speaking of a family of sets, all of them similar in some important ways. The things in R2 look like pairs of real numbers: (3, 4), or (4π, -2e), or (2038, 0.010010001), pairs like that. The things in R3 are triplets of real numbers: (3, 4, 5), or (4π, -2e, 1 + 1/π). The things in R4 are quartets of real numbers: (3, 4, 5, 12) or (4π, -2e, 1 + 1/π, -6) or so. The things in R10 are probably clear enough to not need listing.
It’s possible to add together two things in Rn. At least if they come from the same Rn; you can’t add a pair of numbers to a quartet of numbers, not if you’re being honest. The addition rule is just what you’d come up with if you didn’t know enough mathematics to be devious, though: add the first number of the first thing to the first number of the second thing, and that’s the first number of the sum. Add the second number of the first thing to the second number of the second thing, and that’s the second number of the sum. Add the third number of the first thing to the third number of the second thing, and that’s the third number of the sum. Keep on like this until you run out of numbers in each thing. It’s possible you already have.
You can’t multiply together two things in Rn, though, unless your n is 1. (There may be some conceptual difference between R1 and plain old R. But I don’t recall seeing a mathematician being interested in the difference except when she’s studying the philosophy of mathematics.) The obvious multiplication scheme — multiply matching numbers, like you do with addition — produces something that doesn’t work enough like multiplication to be interesting. It’s possible for some n’s to work out schemes that act like multiplication enough to be interesting, but for the most part we don’t need them.
What we will do, though, is multiply something in Rn by a single real number. That real number is called a “scalar”. You do the multiplication, again, like you’d do if you were too new to mathematics to be clever. Multiply the first number in your thing by the scalar, and that’s the first number in your product. Multiply the second number in your thing by the scalar, and that’s the second number in your product. Multiply the third number in your thing by the scalar, and that’s the third number in your product. Carry on like this until you run out of numbers, and then stop. Usually good advice.
That you can add together two things from Rn, and you can multiply anything in Rn by a scalar, makes this a “vector space”. (There are some more requirements, but they amount to addition and multiplication working like you’d expect.) The term means about what you think; a “space” is a … well … something that acts mathematically like ordinary everyday space works. A “vector space” is a space where the things inside it are vectors. Vectors are a combination of a direction and a distance in that direction. They’re very well-represented as n-tuples. They get represented as n-tuples so often it’s easy to forget that’s just a convenient way to write them down.
This vector space property of Rn makes it a really useful set. R2 corresponds naturally to “the points on a flat surface”. R3 corresponds naturally to an idea of “all the points in normal everyday space where something could be”. Or, if you like, it can represent “the speed and direction something is travelling in”. Or the direction and amount of its acceleration, for that matter.
Because of these mathematicians will often call Rn the “n-dimensional Euclidean space”. The n is about how many components there are in an element of the set. The “space” tells us it’s a space. “Euclidean” tells us that it looks and works like, well, Euclidean geometry. We can talk about the distance between points and use the ideas we had from plane or solid geometry. We can talk about angles and areas and volumes similarly. We can do this so much we might say “n-dimensional space” as if there weren’t anything but Euclidean spaces out there.
And this is useful for more than describing where something happens to be. A great number of physics problems find it convenient to study the position and the velocity of a number of particles which interact. If we have N particles, then, and we’re in a three-dimensional space, and we’re keeping track of positions and velocities for each of them, then we can describe where everything is and how everything is moving as one element in the space R6N. We can describe movement in time as a function that has a domain of R6N and a range of R6N, and see the progression of time as tracing out a path in that space.
We can’t draw that, obviously, and I’d look skeptically at people who say they can visualize it. What we usually draw is a little enclosed space that’s either a rectangle or a blob, and draw out lines — “trajectories” — inside that. The different spots along the trajectory correspond to all the positions and velocities of all the particles in the system at different times.
Though that’s a fantastic use, it’s not the only one. It’s not required, for example, that a function have the same Rn as both domain and range. It can have different sets. If we want to be clear that the domain and range can be of different sizes, it’s common to call one Rn and the other Rm if we aren’t interested in pinning down just which spaces they are.
But, for example, a perfectly legitimate function would have a domain of R3 and a range of R1, the reals. There’s even an obvious, common one: return the size, the magnitude, of whatever the vector in the domain is. Or we might take as domain R4, and the range R2, following the rule “match an element in the domain to an element in the range that has the same first and third components”. That kind of function is called a “projection”, as it gives what might look like the shadow of the original thing in a smaller space.
If we wanted to go the other way, from R2 to R4 as an example, we could. Here set the rule “match an element in the domain to an element in the range which has the same first and second components, and has ‘3’ and ‘4’ as the third and fourth components”. That’s an “embedding”, giving us the idea that we can put a Euclidean space with fewer dimensions into a space with more. The idea comes naturally to anyone who’s seen a cartoon where a character leaps off the screen and interacts with the real world.