Intro to matrices

what I want to do in this video is explore the notion of a matrix outside of the context of a surprisingly good movie that involves Keanu Reeves and it's actually the first of three I guess we could call the three movies combined the matrices and 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 and then the connection really is is that matrices are used a lot when you are simulating things or when you're constructing things in computer science especially in frankly computer graphics so the super intelligent robots that made the the matrix in the movie matrix we're probably using matrices in order to do it if they actually did exist now what is a matrix then well that's a fairly simple answer it's just a rectangular array of numbers so for example this right over here if I have 1 0 negative 7pi 5 and I don't know 11 this is a matrix this is a matrix where 1 0 negative 7pi each of those are an entry in the matrix this matrix right over here has two rows two rows and it has three three columns and because it has two rows and 3 columns people will often say that this is a 2 by 3 matrix whenever they say it's a something by something matrix they're telling you that it has two rows so you see the two rows right over there and they are telling you and they're telling you that it has three columns you see the three columns right over there I could give you other examples of a matrix so I could have I can have a one by one matrix so I could have the matrix one this right over here is a one by one matrix has one row one column I could have a matrix like this 3 7 and 17 what is this well this has one row this is one row we see here and it has three columns this is a one by three matrix I could have a matrix and I think you see where all of this is going figuring out the dimensions of matrix are not too difficult I could matrix that looks like this where it's 3 5 0 0 negative 1 negative 7 this is 7 negative 7 this right over here has 3 rows so it's 3 rows and it has 2 2 columns so we would call this a 3 by 2 let me do that in that same color we would call it a 3 a 3 by 2 matrix 3 rows and 2 columns so fair enough you know that a matrix is just a rectangular 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 those entries but what are matrices good for I still might not be clear what the connection is between this and this right over here and at the most fundamental level these are just ways this is just a compact representation of a bunch of numbers it's a way of representing information they become very valuable in computer graphics because these numbers could represent these numbers could represent the color intensity at a certain point they could represent whether an object is there at a certain point and as we develop an algebra around matrices and when we talk about developing an algebra around matrices we're going to talk about operations that we're going to perform on matrices that we would normally perform with numbers so we're going to learn how we're going to essentially define how to multiply matrices how to add matrices how to even take will learn about taking or an inverse of a matrix and by coming up with an algebra of 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 economic simulation or probability simulation to say oh I have this matrix that represents where different particles are in space or I have this matrix that represents the state of some type of some type of a game and I know the algebra of matrices 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 can actually come up with useful results so that's all matrices are but as you'll see through this they you we can we can define operations on them and then later on when you take a linear algebra course in college you'll learn a lot more of the of how they can be applied and what you can use them to represent