Pop Video: Abstract Vector Spaces

Video Transcript

I’d like to revisit a deceptively simple question that I asked in the very first video of this series, what are vectors? Is a two-dimensional vector, for example, fundamentally an arrow on a flat plane that we can describe with coordinates for convenience, or is it fundamentally that pair of real numbers which is just nicely visualised as an arrow on a flat plane, or are both of these just manifestations of something deeper?

On the one hand, defining vectors as primarily being a list of numbers feels clear-cut and unambiguous. It makes things like four-dimensional vectors or one-hundred-dimensional vectors sound like real, concrete ideas that you can actually work with. When otherwise, an idea like four dimensions is just a vague, geometric notion that’s difficult to describe without waving your hands a bit. But on the other hand, a common sensation for those who actually work with linear algebra, especially as you get more fluent with changing your basis, is that you’re dealing with a space that exists independently from the coordinates that you give it and that coordinates are actually somewhat arbitrary, depending on what you happen to choose as your basis vectors.

Core topics in linear algebra, like determinants and eigenvectors, seem indifferent to your choice of coordinate systems. The determinant tells you how much a transformation scales areas, and eigenvectors are the ones that stay on their own span during a transformation, but both of these properties are inherently spatial, and you can freely change your coordinate system without changing the underlying values of either one. But, if vectors are not fundamentally lists of real numbers, and if their underlying essence is something more spatial, that just begs the question of what mathematicians mean when they use a word like space or spatial.

To build up to where this is going, I’d actually like to spend the bulk of this video talking about something which is neither an arrow nor a list of numbers but also has vector-ish qualities, functions. You see, there’s a sense in which functions are actually just another type of vector. In the same way that you can add two vectors together, there’s also a sensible notion for adding two functions, 𝑓 and 𝑔, to get a new function, 𝑓 plus 𝑔. It’s one of those thing where you kinda already know what it’s gonna be, but actually phrasing it is a mouthful. The output of this new function at any given input, like negative four, is the sum of the outputs of 𝑓 and 𝑔, when you evaluate them each at that same input, negative four. Or, more generally, the value of the sum function at any given input 𝑥 is the sum of the values 𝑓 of 𝑥 plus 𝑔 of 𝑥.

This is pretty similar to adding vectors coordinate by coordinate; it’s just that there are, in a sense, infinitely many coordinates to deal with. Similarly, there’s a sensible notion for scaling a function by a real number, just scale all of the outputs by that number. And again, this is analogous to scaling a vector coordinate by coordinate; it just feels like there’s infinitely many coordinates. Now, given that the only thing vectors can really do is get added together or scaled, it feels like we should be able to take the same useful constructs and problem solving techniques of linear algebra, that were originally thought about in the context of arrows in space, and apply them to functions as well.

For example, there’s a perfectly reasonable notion of a linear transformation for functions, something that takes in one function and turns it into another. One familiar example comes from calculus, the derivative. It’s something which transforms one function into another function. Sometimes in this context, you’ll hear these called operators instead of transformations, but the meaning is the same. A natural question you might wanna ask is what it means for a transformation of functions to be linear. The formal definition of linearity is relatively abstract and symbolically driven compared to the way that I first talked about it in chapter three of this series, but the reward of abstractness is that we’ll get something general enough to apply to functions, as well as arrows.

A transformation is linear if it satisfies two properties, commonly called additivity and scaling. Additivity means that if you add two vectors, 𝐕 and 𝐖, then apply a transformation to their sum, you get the same result as if you added the transformed versions of 𝐕 and 𝐖. The scaling property is that when you scale a vector 𝐕 by some number then apply the transformation, you get the same ultimate vector as if you scale the transformed version of 𝐕 by that same amount. The way you’ll often hear this described is that linear transformations preserve the operations of vector addition and scalar multiplication. The idea of gridlines remaining parallel and evenly spaced is that I’ve talked about in past videos is really just an illustration of what these two properties mean in the specific case of points in 2D space.

One of the most important consequences of these properties which makes matrix-vector multiplication possible is that a linear transformation is completely described by where it takes the basis vectors. Since any vector can be expressed by scaling and adding the basis vectors in some way, finding the transformed version of a vector comes down to scaling and adding the transformed versions of the basis vectors in that same way. As you’ll see in just a moment, this is as true for functions as it is for arrows. For example, calculus students are always using the fact that the derivative is additive and has the scaling property, even if they haven’t heard it phrased that way. If you add two functions then take the derivative, it’s the same as first taking the derivative of each one separately then adding the result. Similarly, if you scale a function, then take the derivative, it’s the same as first taking the derivative, then scaling the result.

To really drill in the parallel, let’s see what it might look like to describe the derivative with a matrix. This will be a little tricky since function spaces have a tendency to be infinite-dimensional, but I think this exercise is actually quite satisfying. Let’s limit ourselves to polynomials, things like 𝑥 squared plus three 𝑥 plus five or four 𝑥 to the seventh minus five 𝑥 squared. Each of the polynomials in our space will only have finitely many terms, but the full space is going to include polynomials with arbitrarily large degree. The first thing we need to do is give coordinates to this space, which requires choosing a basis. Since polynomials are already written down as the sum of scaled powers of the variable 𝑥, it’s pretty natural to just choose pure powers of 𝑥 as the basis function. In other words, our first basis function will be the constant function, 𝑏 zero of 𝑥 equals one. The second basis function will be 𝑏 one of 𝑥 equals 𝑥, then 𝑏 two of 𝑥 equals 𝑥 squared, then 𝑏 three of 𝑥 equals 𝑥 cubed, and so on. The role that these basis functions serve will be similar to the roles of 𝑖-hat, 𝑗-hat, and 𝑘-hat in the world of vectors as arrows.

Since our polynomials can have arbitrarily large degree, this set of basis functions is infinite. But that’s okay, it just means that when we treat our polynomials as vectors, they’re going to have infinitely many coordinates. A polynomial like 𝑥 squared plus three 𝑥 plus five, for example, would be described with the coordinates five, three, one, then infinitely many zeros. You’d read this as saying it’s five times the first basis function plus three times that second basis function plus one times the third basis function, and then none of the other basis functions should be added from that point on. The polynomial four 𝑥 to the seventh minus five 𝑥 squared would have the coordinates zero, zero, negative five, zero, zero, zero, zero, four, then an infinite string of zeros. In general, since every individual polynomial has only finitely many terms, its coordinates will be some finite string of numbers with an infinite tail of zeros.

In this coordinate system, the derivative is described with an infinite matrix that’s mostly full of zeros, but which has the positive integers counting down on this offset diagonal. I’ll talk about how you could find this matrix in just a moment, but the best way to get a feel for it is to just watch it in action. Take the coordinates representing the polynomial 𝑥 cubed plus five 𝑥 squared plus four 𝑥 plus five, then put those coordinates on the right of the matrix. The only term which contributes to the first coordinate of the result is one times four, which means the constant term in the result will be four. This corresponds to the fact that the derivative of four 𝑥 is the constant four. The only term contributing to the second coordinate of the matrix-vector product is two times five, which means the coefficient in front of 𝑥 in the derivative is ten. That one corresponds to the derivative of five 𝑥 squared. Similarly, the third coordinate in the matrix-vector product comes down to taking three times one. This one corresponds to the derivative of 𝑥 cubed being three 𝑥 squared. And after that, it’ll be nothing but zeros. What makes this possible is that the derivative is linear. And for those of you who like to pause and ponder, you could construct this matrix by taking the derivative of each basis function and putting the coordinates of the results in each column.

So, surprisingly, matrix-vector multiplication and taking a derivative, which at first seem like completely different animals, are both just really members of the same family. In fact, most of the concepts I’ve talked about in this series with respect to vectors as arrows in space, things like the dot product or eigenvectors, have direct analogues in the world of functions. Though sometimes they go by different names, things like inner product or eigenfunction. So, back to the question of what is a vector. The point I wanna make here is that there are lots of vector-ish things in maths. As long as you’re dealing with a set of objects where there’s a reasonable notion of scaling and adding, whether that’s a set of arrows in space, lists of numbers, functions, or whatever other crazy thing you choose to define, all of the tools developed in linear algebra regarding vectors, linear transformations, and all that stuff, should be able to apply.

Take a moment to imagine yourself right now as a mathematician developing the theory of linear algebra. You want all of the definitions and discoveries of your work to apply to all of the vector-ish things in full generality, not just to one specific case. These sets of vector-ish things, like arrows or lists of numbers or functions, are called vector spaces, and what you as the mathematician might want to do is say, “Hey everyone! I don’t wanna think about all the different types of crazy vector spaces that you all might come up with.” So what you do is establish a list of rules that vector addition and scaling have to abide by. These rules are called axioms, and in the modern theory of linear algebra, there are eight axioms that any vector space must satisfy if all of the theory and constructs that we’ve discovered are going to apply.

I’ll leave them on the screen here for anyone who wants to pause and ponder, but basically it’s just a checklist to make sure that the notions of vector addition and scalar multiplication do the things that you’d expect them to do. These axioms are not so much fundamental rules of nature as they are an interface between you, the mathematician discovering results, and other people who might want to apply those results to new sorts of vectors spaces. If, for example, someone defines some crazy type of vector space, like the set of all 𝜋 creatures, with some definition of adding and scaling 𝜋 creatures, these axioms are like a checklist of things that they need to verify about their definitions before they can start applying the results of linear algebra.

And you, as the mathematician, never have to think about all the possible crazy vector spaces people might define. You just have to prove your results in terms of these axioms so anyone whose definitions satisfy those axioms can happily apply you results, even if you never thought about their situation. As a consequence, you’d tend to phrase all of your results pretty abstractly, which is to say, only in terms of these axioms, rather than centring on a specific type of vector, like arrows in space or functions. For example, this is why just about every textbook you’ll find will define linear transformations in terms of additivity and scaling, rather than talking about gridlines remaining parallel and evenly spaced, even though the latter is more intuitive, and at least in my view, more helpful for first time learners, even if it is specific to one situation.

So the mathematician’s answer to “what are vectors?” is to just ignore the question. In the modern theory, the form that vectors take doesn’t really matter, arrows, lists of numbers, functions, 𝜋 creatures, really it can be anything so long as there is some notion of adding and scaling vectors that follows these rules. It’s like asking what the number three really is. Whenever it comes up concretely, it’s in the context of some triplet of things. But in maths, it’s treated as an abstraction for all possible triplets of things and lets you reason about all possible triplets using a single idea. Same goes with vectors, which have many embodiments, but maths abstracts them all into a single, intangible notion of a vector space.

But, as anyone watching this series knows, I think it’s better to begin reasoning about vectors in a concrete visualizable setting, like 2D space with arrows rooted at the origin. But as you learn more linear algebra, know that these tools apply much more generally and that this is the underlying reason why textbooks and lectures tend to be phrased, well, abstractly. So with that folks, I think I’ll call it an end to this essence of linear algebra series. If you’ve watched and understood the videos, I really do believe that you have a solid foundation in the underlying intuitions of linear algebra. This is not the same thing as learning the full topic, of course, that’s something that can only really come from working through problems, but the learning you do moving forward can be substantially more efficient if you have all the right intuitions in place. So, have fun applying those intuitions and best of luck with your future learning.