A vector’s a thing you can multiply by a number and then add to another vector.
Oh, I know what you’re thinking. Wasn’t a vector one of those things that points somewhere? A direction and a length in that direction? (Maybe dressed up in more formal language. I’m glad to see that apparently New Jersey Tech’s student newspaper is still The Vector and still uses the motto “With Magnitude And Direction’.) Yeah, that’s how we’re always introduced to it. Pointing to stuff is a good introduction to vectors. Nearly everyone finds their way around places. And it’s a good learning model, to learn how to multiply vectors by numbers and to add vectors together.
But thinking too much about directions, either in real-world three-dimensional space, or in the two-dimensional space of the thing we’re writing notes on, can be limiting. We can get too hung up on a particular representation of a vector. Usually that’s an ordered set of numbers. That’s all right as far as it goes, but why limit ourselves? A particular representation can be easy to understand, but as the scary people in the philosophy department have been pointing out for 26 centuries now, a particular example of a thing and the thing are not identical.
And if we look at vectors as “things we can multiply by a number, then add another vector to”, then we see something grand. We see a commonality in many different kinds of things. We can do this multiply-and-add with those things that point somewhere. Call those coordinates. But we can also do this with matrices, grids of numbers or other stuff it’s convenient to have. We can also do this with ordinary old numbers. (Think about it.) We can do this with polynomials. We can do this with sets of linear equations. We can do this with functions, as long as they’re defined for compatible domains. We can even do this with differential equations. We can see a unity in things that seem, at first, to have nothing to do with one another.
We call these collections of things “vector spaces”. It’s a space much like the space you happen to exist in is. Adding two things in the space together is much like moving from one place to another, then moving again. You can’t get out of the space. Multiplying a thing in the space by a real number is like going in one direction a short or a long or whatever great distance you want. Again you can’t get out of the space. This is called “being closed”.
(I know, you may be wondering if it isn’t question-begging to say a vector is a thing in a vector space, which is made up of vectors. It isn’t. We define a vector space as a set of things that satisfy a certain group of rules. The things in that set are the vectors.)
Vector spaces are nice things. They work much like ordinary space does. We can bring many of the ideas we know from spatial awareness to vector spaces. For example, we can usually define a “length” of things. And something that works like the “angle” between things. We can define bases, breaking down a particular element into a combination of standard reference elements. This helps us solve problems, by finding ways they’re shadows of things we already know how to solve. And it doesn’t take much to satisfy the rules of being a vector space. I think mathematicians studying new groups of objects look instinctively for how we might organize them into a vector space.
We can organize them further. A vector space that satisfies some rules about sequences of terms, and that has a “norm” which is pretty much a size, becomes a Banach space. It works a little more like ordinary three-dimensional space. A Banach space that has a norm defined by a certain common method is a Hilbert space. These work even more like ordinary space, but they don’t need anything in common with it. For example, the functions that describe quantum mechanics are in a Hilbert space. There’s a thing called a Sobolev Space, a kind of vector space that also meets criteria I forget, but the name has stuck with me for decades because it is so wonderfully assonant.
I mentioned how vectors are stuff you can multiply by numbers, and add to other vectors. That’s true, but it’s a little limiting. The thing we multiply a vector by is called a scalar. And the scalar is a number — real or complex-valued — so often it’s easy to think that’s the default. But it doesn’t have to be. The scalar just has to be an element of some field. A ‘field’ is a ring that you can do addition, multiplication, and division on. So numbers are the obvious choice. They’re not the only ones, though. The scalar has to be able to multiply with the vector, since otherwise the entire concept collapses into gibberish. But we wouldn’t go looking among the gibberish except to be funny anyway.
The idea of the ‘vector’ is straightforward and powerful. So we see it all over a wide swath of mathematics. It’s one of the things that shapes how we expect mathematics to look.