We've already dealt with linear equations, linear systems, and matrices. Now let's deal with another big concept in linear algebra, which is vectors. For this course, when we talk about vectors, we will specifically be talking about column vectors. There are other vectors called row vectors. If we talk about row vectors, I will explicitly say row vectors. If I just say vectors or if I say column vectors, both of those are talking about column vectors. What is a column vector? Essentially, what a column vector is, is a matrix with one column. That is it. As simple as that. Everything that has to do with a column vector follows the same rules as matrix algebra. If I have a column vector with n rows here, just like in a matrix, I need the same dimensions with two different matrices if I want to add them. In this case, if I wanted to add this column vector of x_1 through x_n, I would need another vector, also n rows, to add to it. This would hypothetically have an x_2 in it as well. This would be x_1, x_2, x_3, all the way down to x_n, and this would be y_1, y_2, y_3, all the way down to y_n. If you add them just like in a matrix, you would have x_1 plus y_1, x_2 plus y_2, x_3 plus y_3, all the way down to x_n plus y_n. Let's look at what this looks like in a coordinate system. Vectors in general are used very heavily in applications. In physics, a vector might be used for something like velocity. In engineering or certain types of applied math, you might see them in other applications. They're very common. They are definitely used. Understanding how to use them and also what they look like in a coordinate system is pretty important. Let's take a look at this coordinate system. I'm going to put x-axis here, y-axis here. I don't have to do that. You can have any axis you want. Eventually, we'll get to three-dimensional, n-dimensional vectors, in which case, it would be very difficult to visualize something that is more than three dimensions. We might get past that. We'll eventually deal with hyperplanes, but right now, we're just dealing with vectors, and let's do an example with two dimensions. Let's say I have a vector a and a vector b. X is the first entry, y is the second entry of my vectors. In this case, let's do negative 2, 1. My x is negative 2, my y is 1. Negative 2, 1, I put a dot here. Some people will prefer to only use a dot when they deal with coordinate systems and vectors. I would say most people like to deal with arrows, especially when they get into applications like velocity, it would be an arrow. That's a speed and a direction. Let's do b as 4 and 3. 1, 2, 3, 4 in my x direction, and 3 in my y, I have a dot here, an arrow to it. That's what a vector is in a coordinate system. We're just taking for the x variable, for the y variable, we're plotting it on an x, y-axis. Again, I can name this x, y, q, and z, and then I would have a q, z-axis in order to visualize it in a nice manner. We only add these two things. We get, 4 minus 2 is 2, 3 plus 1 is 4. We get 2, 4. 1, 2, 1, 2, 3, 4, you'd have something like this when we add them. There's something to visualize this if you don't want to plot it, it's called the parallelogram rule. It says that if you have two vectors, especially visualized in a coordinate system like we have here, and you wanted to add them, instead of plotting this new point here by doing the actual algebra, you could instead reposition either of the vectors in your favor. No matter which way you do it, you're going to be correct. I can take this vector, let's call it negative 2, 1 is a and b. I can take this b vector, and simply shift it up. A vector can be starting anywhere really, it just says, from where I start, I go in this direction. I made both of these vectors starts at the origin, but you don't have to. It just says from where I start, I'm going to a certain place. I can shift this b to start at where a is. Instead of going from the origin to here, it will go from a to here. If I wanted to do the other side of the coin, if I kept b where it was and I said, well, I want to use a, but instead of starting at the origin, I'm going to start at b, I'm just going to shift it up. It would be something like that. This is vector algebra. We have a column vector or vector and another vector and we're adding them, were' getting a resulting number. But what that means in the coordinate space is, if you start at the origin, and you have both vectors, you take either one, and you move it starting point to the ending point of the other one, wherever that series ends up, is going to be this right here. I have my original a, b. If I shift a to here, I'll now get this, and I'll end up right here. If I start with my original a and then I move my b here, I'll also end up there. That's the parallelogram rule and it's really just, let's take vector algebra and move it to a coordinate system, doesn't make sense.
