 So we've had this introductory look at vectors and matrices and we delved in a slightly deeper into matrices. Let's delve slightly in deeper into vectors and we'll keep on going, we'll keep on going. And the thing I want to talk to you about is vector spaces and vector subspaces. For a moment though, I want you just to ignore the word vector. And I just want us to concentrate on spaces and subspaces. So spaces, imagine I take all the real numbers in the plane. So I only have two axes, x and y, any point there can be written as this x and y, this column vector. So that whole plane, the Cartesian plane and the x and y coordinate system, they make up a space. And we call it a vector space because any point in there, from the origin to that point, is a vector. And that's a vector space. I can bring up three. Remember when we do this, a superscript notation and that would be three-dimensional space. And I just, before we get to that, let's just carry on, I can have R4 as well. And that just means I have this, and usually when it gets larger, we'll rather write x sub 1, x sub 2, x sub 3 and x sub 4. That will be this arbitrary vector in four, four space. So we use these terms sometimes, so that will be four space, four-dimensional space, four space. We don't want to use dimension because that refers just to space itself. Because I want to show you, and that's why I don't want you to use the word vectors, just the spaces. Because if I take all matrices that are two by two matrices, that is a space. I'm not talking about the column view of these as column vectors. I'm talking about the whole matrix that can be seen as a space, a vector space. We put the word vector there. If I look at all functions, all functions, if they can be trigonometric functions, polynomial functions. It doesn't matter, they have function. All the functions form a space. And something that forms a space, obey these eight properties. If I have something and they obey these eight properties and we'll get to them, they form a space. And a space is made up of subspaces. Now the first subspace that you might make a mistake with, so just be aware, is R2 is not a subspace of R3. Now a subspace is, as I said, it's any part of that space that will still have these properties hold. So if I were just to look at, let's just do three-dimensional space. Three-dimensional space X, Y and Z. And they're mutually orthogonal there. So there's the Z pointing out towards you as opposed to the X pointing towards you as opposed to, in the board, they can't draw in a flat board three-dimensional space properly. Not to mention form five-dimensional space, but this notation makes it very simple to deal with very large dimensional spaces. If I take any plane, so a flat sheet of paper and it goes through the origin and I have it anywhere in space, it's got to go through the origin otherwise some of these properties won't hold. So it's got to go through 0, 0, 0. Then that is a subspace. And if I have a vector in that subspace and another vector in that subspace, if I add them, the resultant will still be in that subspace, which is the addition is one of those properties. That will still be in one of those subspaces. Just one, as I say, one little trick. Remember, the two-dimensional, which in this instance is this flat part line here, the Cartesian plane is not a subspace of three-dimensional space because a vector in two-dimensional space will look like this. The plane in three-dimensional space would look like this, X, Y, 0. It has three components. So the plane in the Cartesian XY plane is not a subspace of R3. Just remember that. And neither of R4 or whatever, because that will be another 0. A subspace of this might be all the diagonal matrices, where I have values there, but these values are all 0, so along the main diagonal. Those would all be a subspace. For any values there and there, that will be a subspace of all 2x2 matrices. And if I only look at, say, for instance, polynomials, that will be a subspace of the vector space of all functions. So once again, be wary of that word vector. But I think you get a good idea of what a space is and what a subspace is. So I have something, and I want to see if it is a vector space. And if it is a vector space, I write this with this capital V. And I'm going to... So that's an arbitrary one. It can be this, it can be matrices, it can be polynomials. It doesn't matter what it is. And I'm going to take two arbitrary vectors in that vector space. So any two polynomials or any two functions, and I have any two 2x2 matrices or whatever, we see that the addition on them, so I take two and I add them to each other, that will commute. So V subscript i, V subscript j, V subscript j, V subscript i. Also, if I have three of them, any arbitrary three, and I add them, there's this associative property under addition. There's also this zero vector, such as if I add it to any of the vectors in my vector space, it'll be same as V plus zero, and it will still be V. Which brings us to a very important space, which is actually just this zero vector space, which is just zero. And all of these just being zero. All of those just being zero. Or a function that says zero equals zero. It's all just zeros. And together with that, I suppose, goes eight here, that there's this unique inverse element, that if under addition, I land up with the zero vector. That's easy enough to see. And then there's the scalar, which is one, and this is where multiplication comes in. One times that vector, scalar times that vector, and the scalar being one, just gives me back the vector. If I take two arbitrary scalars, C1 and C2, and multiply them, and then multiply that by the vector, that would be the same as taking one of them, multiplying by the vector, and taking that product, and multiplying it by the scalar as well. And there's also that association, there's also commutivity, because I can change these two around. Then there's this arbitrary scalar C, such as if I add two of them, and then multiply by C would be to multiply C by each one of them, and adding that product. And then if I have two arbitrary scalars, C sub one, C sub two, I add them and then multiply would be the same as the additional multiplication and then the additions. So for any of these, remember a few things. That's not the subspace of that. That's a type of vector space. That's a type of vector space. That's a type of vector space. There's the subspace. The subspace for here at least must go through zero. And very importantly, that we have these eight properties that must hold to make something a vector space.