 Welcome back to our lecture series linear algebra done openly. As usual, I'm your professor today, Dr. Andrew Misseldine. This is the first video in section 2.2 in our textbook called Matrix Equations. This is going to be a continuation of the vector equations we learned about previously, but before we actually talk about these matrix equations, what we're going to do in this video is actually introduce the idea of a matrix product. Now we're not going to do the full blown out matrix multiplication that we'll see in chapter three. At the moment, we're only going to be concerned with the product of a matrix with a vector. Now we've used so far matrices for one purpose and one purpose only. Matrices, particularly the augmented matrix, is a way of encoding a linear system, which we saw in the previous section. We could also write that same linear system as a vector equation. We want to kind of connect this idea together. And so given a matrix, let's say we have some matrix capital A and it's M by N. Remember, we always write the rows first and then the columns when we talk about a matrix here. So we have M rows and N columns. And let's take a vector which lives in FN. So remember that the elements of FN are going to be column vectors within entries. So the number of entries inside the vector X here is the same number of columns that the matrix has. That's an important thing. Otherwise, the following definition will make sense. So whenever you have a matrix, we could talk about the column vectors of the matrix. So like when you have some matrix, one, two, three, four, five, six, something like that, you could think of the matrix itself as a list of vectors. And often like in computer science, that's exactly how you encode matrices there. You have like an array which represents the vector. And then a matrix in some regard is just an array of arrays because you have these column vectors like the first one, the second one, the third one. Sometimes it's also of interest to talk about the row vectors of a matrix. And so because of that, we might want to distinguish vertically versus horizontally. This is the main reason why we write vectors so commonly as column vectors. We want to think of them as the columns of some matrix right here. So let me clean that out. So given a matrix A, the column vectors, let's say that they're A1, A2, up to An. And this is something we'll commonly do. Matrices will typically be denoted with capital Roman letters like capital A, capital B, capital C. And then the column vectors of said matrix will often denote as vectors. And so they'll be bolded like you see on the screen right here. But then we'll bold the lower case letter of that same, the same one. So lower case As, A1, A2, up to An represent the column vectors of the matrix. And we often will denote this in the following manner. We have a matrix A, then the column vectors A1, A2, up to An will list inside of brackets, brackets we're using to denote matrices, of course. We then write the column vectors of A in their order. So this is the first column, the second column, et cetera. So we can represent a matrix via its column vectors. And so now we define the product between a matrix and a vector, A times X. And the product of A times X will be the linear combination of the column vectors of A, where the coefficients in the associated linear combination will be the entries of X. So let me explain what I mean here. The product A times X, this will be the matrix A, which matrix A has column vectors A1, A2, A3, up to An, because it's a M by N matrix. And then the vector X, since it's an Fn, there will be N numbers in that vector, X1, X2, up to Xn. And so the matrix vector product will then be the linear combination of the column vectors of A, where the coefficients of the scalars in this linear combination are exactly the entries of the vector itself. So a quick examples of this. Let's start off with a two by three matrix A right here. So we get one, two, negative three, we get zero, negative two, five. And then we get a vector X right here, zero, three, two. This is a vector that belongs to R3 right here. Notice the number of entries in the vector is exactly the number of columns of the matrix that's necessary. Let me clean this up. And therefore the matrix vector product, what we'll do is we'll take a combination of the column vectors of the matrix. So we have these three column vectors right here. We're looking for that linear combination, one, zero plus two, negative two, plus negative three, five. Now in that linear combination, the entries of the vector are the coefficients for that linear combination, zero, sorry, three, zero, and two. And then if it becomes a process of simplifying this thing, we end up with three, zero. We're going to get zero, zero. And then we're gonna get negative six, 10. And adding together component-wise, we're gonna get negative three, and we're going to get 10 as the product of the matrix by the vector. Here's another example. This time we're gonna have a three by two matrix times by a vector from R2. Again, we have, well, here we have two column vectors, so we're gonna get a linear combination using these two entries right here. And so you're gonna get three times two, negative six, three, plus five times zero, one, two. And so simplifying with the scalar multiplication, we get six, negative 18, nine, and we're gonna add that to the next one, zero, five, and 10. So this adds up to be, of course, six, negative 13, and 19. And this shows us how we can compute the matrix vector product.