 Now, in this introduction to matrices and vectors, we did addition, multiplication. This is going to be the last of these little introductory videos before we get to do something very exciting and, well, a bit more interesting and more in-depth. And what we're going to do here is the last form of multiplication, where we're going to multiply matrices and vectors with each other. So what we have to do, as this is we had with matrix multiplications, we've got to look at the size of these things. So let's have a matrix A and my matrix A is going to be, let's say, 3 and 2 and 1 and 4. And my vector, my vector, let's just make the vector very simple, 1 and 1. And we notice this is 2 by 2, 2 rows, 2 columns, 2 rows, 1 column. And if we do this multiplication, A, V, we see that we can do that because the column number and the first one and the row number and the second one, they decide and the resultant is going to be 2 by 1, 2 by 1. So we're going to end up with a vector and that's very interesting and that has many applications. This is something that acts on a vector and that makes it a very useful thing. We're going to do exactly the same as we did before with that little way to write it down. As I showed you, 3 and 2 and let's make 1 and 4, we're going to have 1, 1 there. And to get these 2, I need 1, 2. This is going to be a 2 by 1, just as promised, 2 by 1. So it is for this first one, it's just what is in this, up and down, this left to right. So it's 3 times 1, 2 times 1, add those, that gives me 5. 3 times 1 is 3, 2 times 1 is 2, 3 plus 2 is 5 and 1 and 4 is also 5. So my resultant vector is 5, 5. So the matrix times the vector gives me another vector and in this instance, the vector is 5, 5. So that's going to be the last type of multiplication we're going to look at. Now we're going to go back and we're going to re-look at matrices and vectors and delve a bit deeper into them and see what else they can tell us and what else are they made up of, what is in their insides. This is going to tease it out and it's very, very exciting stuff. But first, let's go to Mathematica and to see how easy it then is and I'm sure you can do it by yourself without having to watch any of the rest of this video, but let's just go to Mathematica and I'll show you how easy it is to do this matrix times a vector. So here we are in Mathematica, let's do this matrix vector multiplication. And let's just go absolutely nuts. I'm going to have my matrix a and let's create that 1, 2, 4 and 5. So that's four columns there, my first row, 2, 3, 1, 2. Let's go 2, 0, 1, negative 1 and let's make it 2, 2, 1, 5. What about that? Let's have a look at that. Let's do a matrix form and we see a big 4 by 4 matrix, which means we have to have a 4 by 1 vector because the number of columns is 4. So let's make our vector v and let's just make it 2, 3, 1, 5. There we go. And I have my vector v, let's put that in matrix form, matrix form, there we go. And let's do this. So we're going to put a dot v, Mathematica wants a dot notation between these two. So a dot v, there we go, a dot v, let's have that in matrix form and where there we see our 4 by 1 column vector. So the product of a matrix and a vector, as far as the Wolfram language is concerned, put that dot or the full stop period between your matrix and your vector.