 So after we define things, mathematicians also like to then define operations on those things. What can you do with a vector now that you have it? So let's take a look at some of the basic vector operations. So for right now we'll assume that all of our components and everything is going to be in the set of real numbers, but reality is that this makes sense regardless of what our sets are, as long as the individual operations we're describing are computable. So for example suppose I have a vector u and I have a bunch of components, and for future reference I have some other vector v with its own set of components, then I can define the scalar multiple of a vector c times u, and I'm going to define that as c times each of the components. And again if we're working with real numbers c and u are real numbers and so on, so all of these products have a meaning, otherwise whatever our components come from this product should be assumed to have some sort of meaning. So for example if my vector, if I have a four vector, if I have a one two three four component vector, a four tuple, then I might take a look at two times that vector. Well that's going to be two times each of the individual components. So that's going to be two times two, there's my first component, two times negative one, two times five, two times three, and I can expand those out, that's going to be four, negative two, ten, and six, and there's my vector two u. Well again a useful thing to do is to think about what these mean geometrically, and how should I view this scalar multiple? Well let's think about that. So if a vector v, if we view this as the set of directions we're getting from one point to another, but a spatial location doesn't matter, so here's my vector, I'll call it v1, v2, and that tells me how to get from here to there. Well the space location doesn't make a difference, so I can make our lives a lot easier by assuming that our directions are always going to be from the origin to some point, and that point is going to correspond to the components of the vector. The vector says go v1 horizontally, v2 vertically, and so on, if we have more components that will take us to the point, and so the coordinates of the vector correspond to the terminal point when I start at the origin. So now if I look at what happens when I do that scalar multiplication, both of those coordinates are going to be multiplied by the same constant, and so this initial point here, this new point here, well those will cross far on to the end point of a vector, and the thing to notice is that this is the original vector, but I've stretched it out a little bit, and I've gone a little bit farther in what we can think about as the same direction. We could also say that we have a vector that is parallel to the original vector. So something else we can define, again I have my two vectors u and v, and then I can define the sum u plus or minus v as the sum of the individual components of the two vectors. So one thing to note here is that addition is done on the individual components, and this is a common enough practice when we define something to do the same thing to the individual pieces, we say that it's done component-wise. So, for example, I have my two vectors 2, 1, negative 3, and vector v, 1, 4, negative 3, 5, then well I can't actually add those two vectors. In order to add two vectors, I have to add their corresponding components, and I can add u1 to v1, u2 to v2, u3 to v3, but I'm not allowed to leave off this last component, or equivalently if you look at the definition of addition, I have to have the same number of components in the two vectors. So this sum is undefined. On the other hand, if I have a vector with four components, then I can add v, which has four components, to w, and so v plus w, I'm going to add the individual components, that's 1 plus 1, so there's my vector v1, 4, negative 3, 5, I'm going to add 1, 0, 3, and 5 to each of the individual components, and after all the dust settles, that gives you the vector 2, 4, 0, and 10. Well, let's take a look at the geometry. How do I want to view the sum of the two vectors, v, plus v? Well, again, I can view this vector u as the direction we're getting from a point, from the origin, to some terminal point, so this expression here is both the vector, and it also tells us the terminal point, the point that we end up at. Now, when I add this vector v, I am going to end up at a different point, u1 plus v1, u2 plus v2, and if I think about adding that vector v, well, again, vectors are directions we're getting from one point to another, that says that this vector v goes from this point to that point, so I can draw my picture and my geometric sum, vector u plus vector v, might look something like this. Well, again, if I want to think about this sum as a vector, what have I really done? Well, I've gone from the origin to this terminal point, so I've gone on this shortcut path. So, geometrically, what that suggests is if I want to add two vectors, what I'm going to do is I'm going to put down the first vector and then I'm going to drop that second vector, I'm going to make it start where the second, where the first vector ended, so I'm going to add the vectors by joining them and to beginning. Now, anything we do once, we can do any number of times and in various combinations, so an important idea is what's known as a linear combination of vectors. So, let's take some set of vectors, v1 through vn, and assume that each of these vectors is the same size, each of them has k components. Then, we can form a linear combination of the vectors in our set. Well, that's the sum of scalar products. So, we'll have that notation. This is a summation from 1 through n of things that look like something times the vectors. And so, again, this summation notation, well, it's really a1v1 plus a2v2 and so on, up to anvn. Now, note that any of the aIs can be negative if we're assuming vectors in our n, so any of them can be negative, so that means linear combinations could also include subtractions. So, for example, if a vector is v1 is 13 negative 2, v2 is 113, then I can talk about 3v1 minus 2v2, and this is a linear combination and the vector that it produces. Now, let's take a look at that. So, by scalar multiple, 3v1 minus 2v2, so I'll do the scalar multiple there. That's 3 times those components. That's going to be 3, 9, negative 6. That's 2 times these components. That'll be 2, 2, negative 6. And now, I'll do the subtraction component-wise, 3 minus 2, 9 minus 2, negative 6, minus negative 6, and that gets me my final vector, 1, 7, 0.