 As I mentioned earlier, one of the signs that something is extremely important is the number of different ways we have of talking about what is essentially the same topic. And parameterization is another good example of that. So it's often convenient to calculate the value of some quantity based on the values of some other quantities. So for example, we might talk about the area for a rectangle and that's going to be given by the area is equal to length times width. And so we might say that the area is based on, well, mathematically we say it's a function of the length and width. And also we might say that, well, I can talk about all even numbers. Well, I can describe an even number as E equals 2 times n, where n is in the set of natural numbers. So the even numbers can be computed from, can be parameterized by n. And the idea is that this concept of parameterization, well, we can think about it as a function. We can think about it as a formula that gives us the different values. And in general, if we say that something is parameterized by another thing, then we can compute every value of that first thing from specified values of the second thing. So for example, let's say I want to look at all whole numbers that can be expressed as products of 2 and 3. And it's helpful in mathematics and in life in general as to look at a couple of examples and the important mathematical step is to try and generalize from those examples. So for example, a number that is a product of 2's and 3's, so I might take 2 to the 5th, 3 to the 4th, or maybe I'll take something like 2 to the 3rd, 3 to the 2nd, or 2 to the 10th, 3 to the 5th, 2 to the 11th, and so on and so forth, and my generalization, well the thing I notice is that all of these numbers look like 2 to the m, 3 to the n, where m and n are in the set of natural numbers. And so here's my parameterization of these numbers. Now notice the parameterization consists of two parts. One is the actual formula, but then I have to tell you where those parameters m and n are coming from. So that is the whole parameterization, and I do need both pieces to have a complete parameterization. Now let's take a look at another one, find a parameterization of all vectors that can be expressed as linear combinations of my two basic vectors. So it's still helpful to construct a few examples and try to generalize them. So let's take a look at the linear combination, 5 of these and 2 of those. And well that's scalar multiple 15, 5, 20, scalar multiple 2, 8, 10, and a linear combination, I'm going to add those two together. So there's my vector. I could take a different linear combination. And again, after all the dust settles, I have a different combination there. And I could take a general linear combination. So I'll take s of these and t of those, and let's see what happens there. So that's going to be, well, I find the scalar multiples. I add them together, and there's my linear combination. There's my parameterization of all the vectors that I can express as linear combinations of those two base factors. And so I give my parameterization. It's going to look like this. And importantly, again, we specify where those parameters are drawn from. They're both real numbers because we are dealing with linear combinations. One thing we'll find particularly useful is to find the parametric equation of a curve. And so if I have the equation of a curve, I can try to express the points on the curve in terms of a parameter or possibly several parameters. So for example, say I want to parameterize the curve y equals 3x plus 5. So, well, starting out with the examples proved so useful the last time, we should try something different because, well, we don't want to use a winning strategy too often. Well, actually, let's try a few points and see if there's an easy way to find the parameterization. So let's see if x equals 0, y equals 5, and the point that I have is 0, 5. If x equals 3, y equals 14, and I have my point 3, 14. And, well, let's introduce our parameter if x equals t, then y is 3t plus 5. And so my point is t, 3t plus 5. And that suggests I have the parameterization t, 3t plus 5, where t is our x value can be any real number. And so there's my parametric equation for the curve y equals 3x plus 5.