No, not vector in the epidemiological sense. The other, mathy kind of vector. Which, trust me, are fun. At least stick around for the zombies.
This dates back to my college days, when I took Differential Equations. Twice. I’ve always been good with math. Sure, I struggled with plenty of things along the way (percentages, trig identities, multivariable integration. Oooh, and concentrations in chemistry), but DiffEq is where I hit the wall like a coyote hits his own painted-on tunnel. Vector spaces were part of that; an abtruse concept used to justify an abstract concept used to solve some difficult equations that might, in turn, have something to do with the real world. But once I got my head wrapped around them, vector spaces turned out to be a fun and useful bit of math. Hey, it could happen.
Before you get to vector spaces, what the heck is a vector? Simply put, a vector is a combination of a direction and a quantity. You’re driving down the highway; your GPS is telling you that you’re going south at exactly 55 mph (because I know you always scrupulously obey the speed limit.) That’s a vector. Gravity pulls you down. That’s a vector. The direction is straight down, and the quantity is, pick one: The force on you due to gravity, the acceleration due to gravity, or your rapidly increasing speed because you stepped out of a perfectly good airplane.
Okay, that’s all well and good. Going to get a touch more abstract. You remember Cartesian coordinates, right? Let’s say you have a point at
Lucky you! Your shiny point can also be considered as a vector starting at (0, 0) and ending at (7, 7). If you were drawing it out, it’d be an arrow with its base at the origin and the tip of it on your point. If this were a homework assignment, I’d make you figure out its direction (45° or π/4 radians), and it’s magnitude (7√2).
Now, other mortals might be content with their one point, but you’re a man of ambition. With great daring you set out to conquer the Cartesian plane. (7, 6) is the first to fall, and then (8, 5). Soon you seize (4, 7), which we all know is the gateway to (4, 6), and you even begin to dream of securing the fabled origin.
Okay, all those other points? They can be called vectors too. Yes, even (0, 0), although right off the top of my head I’m not sure how you define a direction for a zero-length vector. But never mind that; the set of all these arrows, and indeed all the arrows in your Cartesian plane, is a vector space. Specifically, it’s a two-dimensional vector space. You can quickly make it a three-dimensional vector space by adding height. More dimensions? Hey, I’m not going to be stingy. There was a mathematician, Hilbert, who wanted to work in something so abstract that not even the physicists could find a use for it. He chose infinite-dimensional vector spaces. And then quantum mechanics was all “Hold my beer.”
All that is just dots on a graph; the only thing we’re doing that’s new is drawing the arrow from the origin to the point in question. And up to this point, we’ve only really made up new words for things. So where does it become useful? Zombies.
That’s right; a zombie apocalypse is a vector space. One dimensional, that dimension being the direction of zombies. Seriously. I can mathematically prove it.
There are two fundamental properties a vector space has to satisfy; vector addition, and scalar multiplication. That is, if you have a vector < 3, 7 > you can add another vector to it < 2, -4 > and get < 5, 3 >. And you can take a vector < 2, 8 > and multiply it by a scalar (or what regular people call “a number”), like so: < 2, 8 > * 2.5 = < 5, 20 >.
In the zombie apocalypse vector addition works like so: < 3 zombies > + < 2 zombies > = < 5 zombies >.
Looks like normal addition, doesn’t it? Well, it’s fancy addition because all these quantities are in the direction of zombies.This also works with subtraction, < 3 zombies > – < 2 zombies > = < 1 zombie >.
Scalar multiplication looks like this: 2 * < 7 zombies > = braaaiiinnsss
Okay, not that. But again, this stuff works with division: < 7 zombies > / 2 = < 3.5 zombies >
Whoa whoa whoa, fractional zombies? Half a zombie we can easily picture, crawling towards you with just his hands. A quarter zombie? Two percent of a zombie, crashing through milk jugs? Eh, I’ll let it slide.
What about negative zombies? I can picture < -17 zombies >, but only written on the front of a claymore. Irrational zombies? Of course. Complex zombies? At this point in history, there are only complex zombies. (For those of you missing this joke, well, the mathematicians are cracking up about it. Sorry.)
“Enough of this formalism” I hear you all exclaiming. “What more can you tell us about vector spaces?” (You were shouting that, right? I didn’t mishear?)
Well, to characterize a vector space you need a basis. A basis consists of a set of vectors which define all possible directions in a vector space. Huh? Okay, imagine you’re standing on the point (7, 7). If you want to get to the point (6, 6) you just minus <1, 1> from your current vector (that vector addition thing), and you’re there. And if you want to get to (8,6) you just add the vector <2, 0 > to your current vector. And then you can get to (12,9) by multiplying your current vector by 1.5. Okay? Now let’s say you want to be able to get to any point, but you need to use the minimum number of distinct vectors. (Why? They’re easier to keep track of when you only have a few.) How many vectors do you need?
You need as many vectors to form a basis as you have dimensions. For the Cartesian plane you could choose < 7, 7 > and < 3, 5 > to form your basis. It wouldn’t be a very good choice, but you could choose it. How would you get to, oh, say (-12, 6.4) from there, only adding and subtracting multiples of those vectors? But back to the zombies. Wanna guess what we’re going to use for our one-dimensional basis?
< 1 Zombie >
Okay, let’s say you want a good basis, not one that the other mathematicians are going to laugh at. For that, you need a basis that’s orthogonal (all the vectors are @RightAngles to one another), and normal (that is, all the vectors are one long. One what? One of whatever you’re measuring.) And, because you haven’t gotten enough vocabulary words yet, you can combine these two into one otherwise useless term! You want an orthonormal basis. For a Cartesian plane the usual choices are < 1, 0 > and < 0, 1 > (commonly notated as i and j.) But that’s not the only possible orthonormal basis; you could do < √3/2, 1/2 > and < -1/2, √3/2 >. The other mathematicians won’t laugh, but they will think you’re a pretentious twit.
Right, back to the Zombie Apocalypse. Can we figure out an orthonormal basis for our zombies? You ready for this?
< 1 Zombie >
It has a magnitude of one, and it’s in the direction of zombies, which is perpendicular to all the other dimensions we don’t have. Why mess with perfection?
There are other fun things you can do with this (the vector space has a zero vector at < 0 Zombies >. A great day for mankind.), but let’s look at the other side of the Zombie apocalypse. What about the scrappy human survivors?
Human survivors can be considered a multidimensional vector space. You’ve got your male lead, you’ve got your love interest, you’ve got your nerd, and you’ve got your black comic relief that dies horribly, and your mother who inevitably turns to heighten the drama.So, for a sample vector in this vector space, we could have
<1 male lead, 1 love interest, 4 nerds, 1.5 black guys that die horribly, 2 mothers who inevitably turn to heighten the drama >
Hey, my kindergarten teacher assures me that last one is a thing. Moving along, we can test if this meets the conditions of a vector space; does it allow vector addition and scalar multiplication?
In theory, we could add more of one character type to the movie. You could take that vector up above and add < 0, 1 love interest, 0, 0, 0 > to get two love interests. I’ve yet to experimentally verify this one. Ladies, if any of you would like to advance the cause of science…
Right. Vectors. You can also do scalar multiplication; take a survivor vector and multiply it by two and you get that scene in Shaun of the Dead where they meet the other group of survivors. You can also define a zero vector, that is, the zombies win.
And a basis? Well, we’ve got to determine if our categories are orthogonal to one another. Clearly no amount of “mother who inevitably turns to heighten the drama” can become “male lead”, not unless something funny’s going on. There might be some overlap between the categories of “nerd” and “love interest”, but you weren’t fooled by the prop glasses, were you? Our best bet is to categorize all the possible human survivor roles and determine to the best of our ability which are orthogonal to one another, but that sounds like a lot of work so I haven’t done it.
Normalization, at least, is easy. One nerd provides a unit vector in the nerd direction, thank you very much.
There’s one last trick to this analogy; transformation. A vector space can be transformed into another vector space of equal or lesser dimensionality. What that means is that you can take a multi-dimensional vector space like the survivors, and turn it into a one-dimensional vector space by biting them. You can’t, however, run it in reverse and turn a one-dimensional vector space back into a multi-dimensional vector space. She’s gone, man.
In mathematical terms, I’ve got no idea what all a vector space transformation entails. I just remember the biting. Hey, it’s not a perfect system.
Okay, that’s all fun, but is any of this stuff useful? Yes, actually. Back in my college days, I played entirely too much World of Warcraft. I also did entirely too much math. I figured “This geek thing isn’t getting me any dates. Clearly I’m not geeky enough; let’s knock this up a notch,” and combined the two. I worked out a way to tell which items were better than other items in terms of damage output, without having to worry about whether that damage is coming from hit chance or crit chance or attack power.
That left me with a problem, though. Does my formula work? Can you prove it? Ain’t no textbook where I’m going; can’t just look it up and see. So how do I know whether my operations are valid? And then it hit me; the items in the game satisfy the two conditions of a vector space (scalar multiplication and vector addition), and as such, I was free to use any kind of vector operation, which means that the fancy mathematical tricks I used to complete my calculation are entirely valid.
I’ve had other successes with the concept. It’s helped me organize and visualize data in situations where I’m out of my usual range of experience. I seem to find it easier to work with a six-dimensional vector space than a point in six-dimensional space, possibly because I stop trying to picture it. Now I just need to apply the zombies to the differential equations themselves and I’ll be in business.Published in