Algebra Matrices Basic matrix operations Introduction to the matrix Matrix dimensions Scalar multiplication Scalar matrix multiplication Matrix addition and subtraction Matrix addition and subtraction Transpose of a matrix Matrix transpose Introduction to the matrix Back Introduction to the matrix ⇐ Use this menu to view and help create subtitles for this video in many different languages. You'll probably want to hide YouTube's captions if using these subtitles. What I want to do in this video is to explore the notion of a matrix Outside the context of a surprisingly good movie that involves Keanu Reeves and it is actually the first of three, I guess we could call it three movies combined the Matrixes And in fact there is a relationship between the movie, which is about a virtual reality constructed by super smart computers and the relation and the notion of what a matrix is when you study it in mathematics. Or when you study it in computer science, the connection really is that matrixes are used a lot when you are assimilating things or constructing things in computer science, especially in frankly, computer graphics. Then, the super intelligent robots that made the matrix in the movie Matrix probably using matrixes to in order to do it. If they actually did exist. Now, what is a matrix then? Well, that is a fairly simple answer: its just a rectangular array of numbers. So, for example, this right over here: 1, 0, -7, pi, 5 and ... 11. This is a matrix. This is a matrix where 1, 0, -7, pi ... each of those are entries in the matrix. This matrix right over here has 2 rows and 3 columns. And, because it has 2 rows and 3 columns, people will say that it is a 2 x 3 matrix. Whenever they say ' this is a something by something matrix ', they're telling you that it has 2 rows. So you see the 2 rows right over there and they are telling you And they are telling you that it has has 3 columns, you see the 3 columns right there. I can give you other examples of matrixes. So I could have a 1 x 1 matrix ... So I could have the matrix 1. This right over here is a 1 x1 matrix. It has one row and one column I can have a matrix like this ... 3, 7 and 17. What is this? Well, this has 1 row, this is the one row we see right here. And it has 3 columns, this is a 1 x 3 matrix. I could have a matrix, and I think you see where all of this is going, Figuring out the dimensions of a matrix is not too difficult. I could have a matrix that looks like this, where its 3, 5, 0, 0, -1, -7 This 7 is negative 7. This right over here has 3 rows, so its 3 rows and it has 2 columns. So we would call it a 3 by 2 matrix, let me do that in the same colour, we would call it a 3 by 2 matrix, 3 rows, and 2 columns. So fair enough, you know that a matrix is just a rectangular array of numbers, you can say what its dimensions are, you know that each of these numbers, that take one of these positions, we just call these entries, But what are matrixes good for? I still might not be clear what the connection is between this and this right over here. And the most fundamental level well these are just ways to, a compact representation, of a bunch of numbers, its a way of representing information. They become very valuable in computer graphics because these numbers could represent the colour intensity at a certain point They could represent whether or not an object is there at a certain point. And as we develop an algebra around matrixes, and when we talk about developing an algebra around matrixes, We are going to talk about operations that we are going to perform on matrixes. That we would normally perform with numbers. So we are going to learn how or essentially define how to multiply matrixes, add matrixes, How to even take the inverse of a matrix and coming up with an algebra, how we manipulate these things, It'll become very useful in the future when you're trying to write a computer graphics program, or you're trying to do an economics simulation, or a probability simulation, to say oh, I have this matrix that represents where different particles are in space, or this matrix represents the state of some type of game. And I know the algebra of matrixes, and I know ways of doing it very efficiently, So that I can, multiply a bunch of them, or I could come run a simulation, and I could actually come up with useful results. So that's all matrixes are, but as you'll see through this, we can define operations on them. And later on, when you take a linear algebra course in college, you learn a lot more of the depth, of how they can be applied, and what you can use them to represent. 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