 Let's take a look at a different set and see if it's a vector space. So how about the set of real numbers, and let's see if they form a vector space over the rational's Q. Now notice an important feature here. The rationales are the scalar field over which we're considering a vector space. They're not the same things that produce the components of our vectors themselves. So we could have a difference between the scalar field and the source of the vector components. Well again, we'll submit our membership application and have it reviewed by the membership committee. So again, we want to consider any three elements of R, M, N and P for example, and anything in our scalar field A and B. And we'll check our properties. So the first requirement, we have to have closure under addition. If I take two real numbers, there's some, is in fact a real number. Closure under scalar multiplication. If I take something from the scalar field, Q, and multiply it by one of the vectors, M, then AM is a rational number times a real number, and that's definitely going to be a real number. So we still have closure under scalar multiplication. Commutativity of addition. We want to check that the sum of two real numbers is the same no matter how I add them, and that's certainly true. Associativity of addition. If I add three numbers, I get the same result regardless of how I group them. So far, so good. Our set of real numbers has met four of the requirements to form a vector space over the rational's Q. And so, like baking a cake, once you've thrown in the flour, sugar, and eggs, you're done baking the cake and you can serve it. Well, you probably need to finish the job. So we have to check the other requirements for being a vector space. So we need to confirm that there is actually zero vector, that there's some real number, which if we add it to any other real number, gives us what we started with. And the number zero works perfectly well as that zero plus m is always going to be equal to m. As with the field itself, it's not enough that zero exists. Zero has to be one of the elements of our set. And since zero is a real number, we do have a zero vector. Along with the zero, we must also have the ad-viv inverse. If I have any vector, any real number, then there's going to be some vector minus v, for which the sum is zero. So if I have a real number, I have a negative m, and I know that m plus negative m is going to be zero. As with the zero itself, it's not enough that the additive inverse exists, but it must be available. It must be part of the set. So here we want to check that we do have the additive inverse minus m, and it's available minus m is in fact a real number. And so this set of vectors has the additive inverses. Our next requirement, multiplication by one. If I take one and multiply it by any real number, I do actually get the real number that I started with. Associativity of scalar multiplication, a times bm, is in fact the same thing as ab times m. And we have our two distributive properties of our scalars, which were elements of our set of rational numbers, over our vectors, which were our real numbers. And so a scalar a times the sum m plus n is in fact am plus an. So we have left distributivity, and we'll check right distributivity. If I have a plus b times m, I do actually get am plus bm. And this means that the set of real numbers has met all of the requirements it needs in order to be a vector space over the rationals. And so the membership committee sends a letter to the set of real numbers, congratulating it on its being accepted into the vector spaces over q, and asks whether it would like to pay its dues through direct deposit, or whether it wants to be billed monthly.