 Well, now that we can add and subtract and scalar multiply vectors and also matrices, let's talk about a more general concept known as a vector space. So a set of vectors, form a vector space, provided that all of the following conditions are met for any vectors u, v, and w in our set of vectors and all real numbers a, b, and c, which are parts of r. We typically, we can actually extend the notion of a vector space. So now that we have a little bit of ability to arithmetic with vectors and matrices, we can introduce the concept of what's called a vector space. Now this term vector in vector space should be viewed as any sort of tuple or even any sort of matrix or even later on in mathematics. You might also look at tensors and a whole bunch of other things. The idea is that anything can be, can be included in this concept of a vector space. So we'll begin with the following idea. A set of things, v, is going to form what we call a vector space, provided that we meet all of the following conditions for any of these vectors in our set of vectors and any real numbers a, b, and c. And again, the important thing to remember is that these things here don't actually have to be vectors. They don't have to be n tuples. They could be things like matrices. They could be any of a number of other things. And the only real requirement is that they have to meet all of the following conditions. And what are those conditions? Well, there's a rather long list of conditions. So fair warning, this is a very extensive list. But again, to some extent, the more important, the more valuable something is, the more powerful something is, the more it'll cost you to implement it. Vector spaces are very powerful ideas and it does mean that we have to meet a lot of requirements to be allowed to be called a vector space. So what are those requirements? Well, the first one is we want to have closure under addition. So given two of these elements of the vector space, given two of these elements, I want to make sure that the sum, however I define that sum, is going to be part of my vector space. I also want to have closure under scalar multiplication. Given any vector and any real number, I want that scalar multiple to also be an element of my vector space. Commutivity is nice. I want to make sure that this addition of two vectors, however it's defined, I want to make sure that that gives me the same result, no matter which way I do that addition. The next thing I want is associativity of vector addition. So if I write V plus U plus W, if I do U plus W first and then add V, what I want to do is I want to make sure that it's the same result if I add V and U first and then add W. So there's our associativity of vector addition. I want there to be a zero vector. I want there to be a vector, which I'll write as a zero vector, so that for anything that I choose in my vector space, then that sum adding this to that vector doesn't change anything. And if we have commutativity, V plus zero should be the same as zero plus V, so I don't have to include that as a separate requirement. Well, once you have zero, we also would like there to be an inverse, an additive inverse. So for any vector V, I'd like to find... I'd like to know that there is a vector, which I'll designate minus V, still in my vector space, for which if I add the two vectors together, I get my zero vector back. And again, commutativity of addition back here guarantees that I can go negative V plus V also will be zero. I also want to have this property of scalar multiplication by one. Now, I've defined scalar multiplication here. I assume that scalar multiplication of some sort exists, but I want to make sure that the multiplication by one of any vector does give me what I want it to give me, which is the original vector back. And I want to also have associativity of scalar multiplication. If I take two scalars A and B, A times scalar multiple B times vector should be the same as the result that I get by multiplying A, B together by the vector. And then I want to have two forms of the distributive property. One of them is the distributive property of scalars over vectors. If I have the scalar multiple A over U plus V, I want that to be what I'd like it, what I would expect it to be, AU plus AV. And then the other thing that I have is this distributive property of vectors over scalars. If the same scalar, sorry, if two different scalars are being multiplied by the same vector, I want to be able to combine those two scalars together and get a single product. So here are my 10 requirements for what makes a vector space. And if I fail any of these requirements, we're not a vector space. So in order to be a vector space, we have to meet all 10 of these requirements. Now, this leads us to the concept of proof. And higher mathematics is generally proof-based, and linear algebra is a traditional place that we start to introduce proofs in mathematics. And there are two very important things to keep in mind when you're trying to prove anything. And to paraphrase Rabbi Hillel, definitions are the whole of mathematics. All else is commentary. Now, what does this mean? Well, this means that you really need to know the definitions. You cannot do mathematics without knowing the definitions. Everything else can be derived from the definitions of the rules of logic. The other thing is when you're trying to prove something, you should try as hard as possible to show that what you're trying to prove is, in fact, false. This may seem to be counterintuitive, because if I'm trying to prove something, I believe it's true. So why should I try to prove that it's false? The reason is if it is false and you don't discover it beforehand, somebody else will, and they won't be as nice to you as you might be. So one thing that we do have to talk about is indefinitions. And in ordinary language, we have definitions like a dog is an animal with four legs. And the problem is that in mathematics and in logic, this is not a definition because true definitions have to go both ways. In particular, I say a dog is an animal with four legs. Well, that's true. But on the other hand, I can't reverse it. I can't say that an animal with four legs is a dog. I do not believe that is the case. I say that is false. And because this does not go in both directions, this is not a definition. So what we would say is that it is a property of a dog, but it is not a definition for what a dog is. Now, mathematically, we have a definition for vector space, and that means two things. Anything we call a vector space has all ten of those listed properties. For example, we're going to make the following claim. We're not going to prove it, but the important claim is that the real numbers using the ordinary rules of arithmetic form a vector space. And what that means, because I'm saying that the real numbers using the ordinary rules for arithmetic form a vector space, I'm saying that the real numbers using the ordinary rules for arithmetic have all ten of those nice properties. So if I take two things in the real numbers, and my definition of a vector space assumes I have something drawn from the real numbers, so I take two elements. R here is my vector space. R here is the set of real numbers. I'm allowed to use, I get to have the properties that the sum of any two real numbers is a real numbers. The product of any real number by C is going to be a real number. I have commutativity of addition. A plus B is the same as B plus A. I have associativity of addition. I have a zero vector. I have the additive inverse. I have multiplication by one. And I have my distributive properties. So knowing that I have a vector space, I know a lot about R. This is why being able to determine whether something is a vector space is so important. So going back to our definitions, anything that we call a vector space has all of the listed properties, but also anything that meets all of the listed properties because it's a proper definition. If it meets the listed properties, it's a vector space. And this leads to a fairly important, but relatively simple, but very important type of proof in mathematics, proving that this is that, proving that something is something else. And in this case, we have our definition for vector space, and so what we generally do is we try to show that whatever this is has all of the required properties. And importantly, we have to meet all of the requirements. If we fail even a single one of the requirements, we do not get to call something a vector space. So we'll take a look at that in the next video.