WEBVTT 00:00:01.300 --> 00:00:06.800 Let's learn about matrices. So, what is a, well, what I do I mean when I say matrices? 00:00:06.800 --> 00:00:10.400 Well, matrices is just the plural for matrix. 00:00:10.400 --> 00:00:15.700 Which is probably a word you're familiar with more because of Hollywood than because of mathematics. 00:00:15.700 --> 00:00:20.900 So, what is a matrix? Well, it's actually a pretty simple idea. 00:00:20.900 --> 00:00:24.500 It's just a table of numbers. That's all a matrix is. 00:00:24.500 --> 00:00:27.800 So, let me draw a matrix for you. 00:00:27.800 --> 00:00:30.300 I don't like that toothpaste colored blue, so, let me use another color. 00:00:30.300 --> 00:00:37.600 This is an example of a matrix. If I said, I don't know I'm going to pick some random numbers; 00:00:37.600 --> 00:00:46.000 Five, one, two, three, zero, minus five. That is a matrix. 00:00:46.000 --> 00:00:51.500 And all it is is a table of numbers and, oftentimes if you want to have a variable for a matrix, you 00:00:51.500 --> 00:00:54.600 use a capital letter. So, you could use a capital 'A'. 00:00:54.600 --> 00:01:00.100 Sometimes in some books they make it extra bold. So it could be a bold 'A', would be a matrix. 00:01:00.100 --> 00:01:04.500 And, just a little bit of notation, So, they would call this matrix. Or, we would call 00:01:04.500 --> 00:01:10.100 this matrix, just by convention, you would call this a two by three matrix. 00:01:10.100 --> 00:01:16.500 And, sometimes they actually write it '2 by 3' below the bold letter they use to represent the matrix 00:01:16.500 --> 00:01:18.400 What is two? And, what is three? 00:01:18.400 --> 00:01:23.200 Well, two is the number of rows. We have one row, two row. This is a row, this is a row. 00:01:23.200 --> 00:01:26.300 We have three columns; one, two , three. 00:01:26.300 --> 00:01:28.500 So, that's why it's called a two by three matrix. 00:01:28.500 --> 00:01:34.200 When you say, you know, if I said, if I said that B, I'll put it extra bold. 00:01:34.200 --> 00:01:42.677 If B is a five by two matrix, that means that B would have, I can, let me do one 00:01:42.677 --> 00:01:46.892 I'll just type in numbers; zero, minus five, ten. 00:01:49.300 --> 00:01:52.600 So, it has five rows, it has two columns. 00:01:52.600 --> 00:01:56.000 We'll have another column here. So, let's see; minus ten, three, 00:01:56.000 --> 00:02:04.100 I'm justing putting in random numbers here. Seven, two, pi. 00:02:04.100 --> 00:02:07.000 That is a five by two matrix. 00:02:07.000 --> 00:02:11.700 So, I think you'd now have a kind of a convention that all a matrix is is a 00:02:11.700 --> 00:02:15.000 table of numbers. You can represent it when you're doing it in variable form 00:02:15.000 --> 00:02:19.100 you represent it as bold face capital letter. Sometimes you'd write two by three there. 00:02:19.100 --> 00:02:22.700 And, you can actually reference the terms of the matrix. 00:02:22.700 --> 00:02:26.300 In this example, the top example, where we have matrix A. 00:02:26.300 --> 00:02:32.600 If someone wanted to reference, let's say, this, this element of the matrix. 00:02:32.600 --> 00:02:37.400 So, what is that? That is in the second row. It's in row two. 00:02:37.400 --> 00:02:39.100 And, it's in column two. Right? 00:02:39.100 --> 00:02:42.500 This is column one, this is column two. Row one, row two. 00:02:42.500 --> 00:02:45.100 So, it's in the second row, second column. 00:02:45.100 --> 00:02:51.900 So, sometimes people will write that A, then they'll write, you know 00:02:51.900 --> 00:02:58.500 two comma two is equal to zero. 00:02:58.500 --> 00:03:02.100 Or, they might write, sometimes they'll write a lowercase a, 00:03:02.100 --> 00:03:07.100 two comma two is equal to zero. 00:03:07.100 --> 00:03:11.700 Well, what is A? These are just the same thing. 00:03:11.700 --> 00:03:14.200 I'm just doing this to expose you to the notation, because 00:03:14.200 --> 00:03:16.100 a lot of this really is just notation. 00:03:16.100 --> 00:03:21.800 So, what is a, one comma three? 00:03:21.800 --> 00:03:24.600 Well, that means we're in the first row and the third column. 00:03:24.600 --> 00:03:27.600 First row; one, two, three. It's this value right here. 00:03:27.600 --> 00:03:29.200 So, that equals two. 00:03:29.200 --> 00:03:32.100 So, this is just all notation of what a matrix is; 00:03:32.100 --> 00:03:34.100 it's a table of numbers, it can be represented this way. 00:03:34.100 --> 00:03:37.000 We can represent its different elements that way. 00:03:37.000 --> 00:03:38.300 So, you might be asking 00:03:38.300 --> 00:03:41.600 "Sal, well, that's nice, a table of numbers with fancy 00:03:41.600 --> 00:03:44.200 words and fancy notations. But, what is it good for?" 00:03:44.212 --> 00:03:46.100 And that's the interesting point. 00:03:46.100 --> 00:03:51.600 A matrix is just a data representation. It's just a way of writing down data. 00:03:51.600 --> 00:03:53.600 That's all it is. It's a table of numbers. 00:03:53.600 --> 00:03:57.800 But, it can be used to represent a whole set of phenomenon. 00:03:57.800 --> 00:04:01.500 And if you're doing this in you Algebra 1 or your Algebra 2 class 00:04:01.500 --> 00:04:03.600 you're probably using it to represent linear equations. 00:04:03.600 --> 00:04:07.854 But, we will learn, later, that it, and I'll do a whole set of videos 00:04:07.869 --> 00:04:10.600 on applying matrices to a whole bunch of different things. 00:04:10.600 --> 00:04:14.500 But, it can represent, it's very powerful and if you're doing 00:04:14.500 --> 00:04:19.100 computer graphics, that matrixes...The elements can represent pixels on your screen, 00:04:19.100 --> 00:04:21.400 they can represent points in coordinate space, 00:04:21.400 --> 00:04:23.000 they can represent...Who knows! 00:04:23.000 --> 00:04:24.900 There's tonnes of things that they can represent. 00:04:24.900 --> 00:04:27.600 But, the important thing to realize is that a matrix 00:04:27.600 --> 00:04:30.500 isn't, it's not a natural phenomenon. 00:04:30.500 --> 00:04:34.700 It's not like a lot of the mathematical concepts we've been looking at. 00:04:34.700 --> 00:04:37.700 It's a way to represent a mathematical concept. 00:04:37.700 --> 00:04:40.400 Or, a way of representing values. But you kinda have to 00:04:40.400 --> 00:04:43.000 define what it's representing. 00:04:43.000 --> 00:04:44.700 But, lets put that on the back burner a little bit 00:04:44.700 --> 00:04:48.300 in terms of what it actually represents. 00:04:48.300 --> 00:04:52.200 And the, oh, my wife is here. She's looking for our filing cabinet. 00:04:52.200 --> 00:04:54.500 But anyway, back to what I was doing. 00:04:54.500 --> 00:04:57.100 So, so, lets put on the back burner what a matrix is 00:04:57.100 --> 00:04:59.400 actually representing. Let's learn the conventions. 00:04:59.400 --> 00:05:02.200 Because, I think, uhm, at least initially, that tends to be 00:05:02.200 --> 00:05:04.015 the hardest part, How do you add matrices? 00:05:04.015 --> 00:05:06.408 How do you multiple matrices? How do you invert a matrices? 00:05:06.408 --> 00:05:09.069 How do you find the determinant of a matrix? 00:05:09.069 --> 00:05:11.400 I know all of those words might sound unfamiliar. Unless, 00:05:11.400 --> 00:05:13.700 you've already been confused by then in your algebra class. 00:05:13.700 --> 00:05:15.900 So. I'm gonna teach you all of those things first. 00:05:15.900 --> 00:05:18.400 Which are all really human-defined conventions. 00:05:18.400 --> 00:05:22.700 And then, later on, I'll make a whole bunch of videos on the intuition behind them, 00:05:22.700 --> 00:05:26.700 and what they actually represent. So, let's get started. 00:05:26.700 --> 00:05:29.700 So, lets say I wanted to add these two matrices. 00:05:29.700 --> 00:05:33.600 Let's say, the first one, let me switch colors. Let's say, 00:05:33.600 --> 00:05:37.700 I'll do relatively small ones, just, not to waste space. 00:05:37.700 --> 00:05:42.500 So, you have the matrix; three, negative one, I don't know, 00:05:42.500 --> 00:05:49.100 two, zero. I don't know, let's call that A, capital A. 00:05:49.100 --> 00:05:54.400 And let's say matrix B, and I'm just making up numbers. 00:05:54.400 --> 00:06:06.300 Matrix B is equal to; minus seven, two, three, five. 00:06:06.300 --> 00:06:14.000 So, my question to you is: What is A, 00:06:14.000 --> 00:06:16.300 so I'm doing it bold like they do in the text books, plus 00:06:16.300 --> 00:06:21.700 matrix B? So, I'm adding two matrices. And, once again 00:06:21.700 --> 00:06:25.700 this is just human convention. Someone defined how matrices add. 00:06:25.700 --> 00:06:27.500 They could've defined it some other way. But, they said; 00:06:27.500 --> 00:06:29.846 we're gonna make matrices add the way I'm 00:06:29.846 --> 00:06:32.500 about to show you because it's useful for a whole set of phenomenon. 00:06:32.500 --> 00:06:35.000 So, when you add two matrices you essentially just add 00:06:35.000 --> 00:06:40.000 the corresponding elements. So, how does that work? 00:06:40.000 --> 00:06:43.000 Well, you add the element that's in row one column one with 00:06:43.000 --> 00:06:46.100 the element that's in row one column one. Alright, so, it's 00:06:46.100 --> 00:06:50.500 three plus minus seven. So, three plus minus seven. 00:06:50.500 --> 00:06:55.000 That'll be the one-one element. Then, the row one column two element 00:06:55.000 --> 00:06:58.608 will be minus one plus two. 00:06:58.608 --> 00:07:01.700 Put parenthesis around them so you know that these are 00:07:01.700 --> 00:07:05.400 separate elements. And, you could guess how this keeps going. 00:07:05.400 --> 00:07:20.700 This element will be two plus three. This element, this last element will be zero plus five. 00:07:20.700 --> 00:07:26.700 So, that equals what? Three plus minus seven, that is minus four. 00:07:26.700 --> 00:07:32.000 Minus one plus two, that's one. Two plus three is five. And, 00:07:32.000 --> 00:07:39.800 zero plus five is five. So, there we have it, that is how we humans have defined the addition of two matrices. 00:07:39.800 --> 00:07:43.200 And, by this definition, you can imagine that this is going to be the same thing 00:07:43.200 --> 00:07:49.100 as B plus A. Right? And remember, this is something we have to think about 00:07:49.100 --> 00:07:53.000 because we're not adding numbers anymore. You know one plus two is the same as 00:07:53.000 --> 00:07:56.700 two plus one. Or, any two normal numbers, it doesn't matter what order you 00:07:56.700 --> 00:07:59.900 add them in. But matrices it's not completely obvious. But, when you define it in this way 00:07:59.900 --> 00:08:03.700 it doesn't matter if we do A plus B or B plus A. Right? 00:08:03.700 --> 00:08:06.600 If we did B plus A, this would just say negative seven plus three. 00:08:06.600 --> 00:08:10.100 This would just say two plus negative one. But, it would come out to the same values. 00:08:10.100 --> 00:08:11.900 That is matrix addition. 00:08:11.900 --> 00:08:15.300 And, you can imagine, matrix subtraction, it's essentially the same thing. 00:08:15.300 --> 00:08:21.592 We would...Well, actually let me show you. What would be A minus B? 00:08:27.038 --> 00:08:32.300 Well, you can also view that, this is capital B, it's a matrix 00:08:32.300 --> 00:08:34.800 that's why I'm making it extra bold. But, that's the same thing as; 00:08:34.800 --> 00:08:42.800 A plus minus one, times B. What's B? Well, B is; 00:08:42.800 --> 00:08:47.800 minus seven, two, three, five. And, when you multiply 00:08:47.800 --> 00:08:50.400 a scalar, when you just multiply a number times the matrix, 00:08:50.400 --> 00:08:52.700 you just multiply that number times every one of its elements. 00:08:52.700 --> 00:08:58.400 So, that equals A, matrix A, plus the matrix, we just multiply 00:08:58.400 --> 00:09:02.400 the negative one times every element in here. So, seven, 00:09:02.400 --> 00:09:08.400 minus two, minus three, five. And then we can do 00:09:08.400 --> 00:09:11.700 what we just did up there. We know what A is. So, 00:09:11.700 --> 00:09:15.800 this would equal, let's see, A is up here. So, three plus 00:09:15.800 --> 00:09:21.200 seven is ten, negative one, plus negative two is minus three, 00:09:21.200 --> 00:09:28.900 two plus minus three is minus one and zero plus five is five. 00:09:28.900 --> 00:09:31.600 And, you didn't have to go through this exercise right here. 00:09:31.600 --> 00:09:33.800 You could have, literally, just subtracted these elements from these elements 00:09:33.800 --> 00:09:35.200 and you would have gotten the same value. 00:09:35.200 --> 00:09:38.500 I did this because I wanted to show you also that multiplying 00:09:38.500 --> 00:09:41.300 a scalar times, or just a value or a number, times a matrix 00:09:41.300 --> 00:09:46.600 is just multiplying that number times all of the elements of that matrix. 00:09:46.600 --> 00:09:50.900 And, so what...By this definition of matrix addition what do we know? 00:09:50.900 --> 00:09:54.200 Well, we know that both matrices have to be the same size, 00:09:54.200 --> 00:09:58.700 by this definition of the way we're adding. So, for example 00:09:58.700 --> 00:10:01.100 you could add these two matrices, You could add, I don't know, 00:10:01.100 --> 00:10:08.500 one, two, three, four, five, six, seven, eight, nine to this matrix; 00:10:08.500 --> 00:10:14.500 to, I don't know, minus ten, minus one hundred, minus one thousand. 00:10:14.500 --> 00:10:20.100 I'm making up numbers. One, zero, zero, one ,zero, one. 00:10:20.100 --> 00:10:21.800 You can add these two matrices. Right? 00:10:21.800 --> 00:10:24.900 Because they have the same number of rows and the same number of columns. 00:10:24.900 --> 00:10:30.400 So, for example, if you were to add them. The first term up here would be one plus minus ten, 00:10:30.400 --> 00:10:34.400 so, it would be minus nine. Two plus minus one hundred, minus ninety-eight. 00:10:34.400 --> 00:10:39.500 I think you get the point. You'd have exactly nine elements and you'd have three rows of three columns. 00:10:39.500 --> 00:10:44.800 But, you could not add these two matrices. You could not add... 00:10:44.800 --> 00:10:48.600 Let me do it in a different color, just to show it is different, 00:10:48.600 --> 00:10:52.500 You could not add, this blue, you could not add this matrix; 00:10:52.500 --> 00:11:03.400 minus three, two to the matrix; I don't know, nine, seven. 00:11:03.400 --> 00:11:05.100 And why can you not add them? 00:11:05.100 --> 00:11:07.700 Well, they don't have corresponding elements to add up. 00:11:07.700 --> 00:11:11.600 This is a one row by two column, this is one by two 00:11:11.600 --> 00:11:15.800 and this is two by one. So, they don't have the same dimensions 00:11:15.800 --> 00:11:18.700 so we can't add or subtract these matrices. 00:11:18.700 --> 00:11:22.300 And, just as a side note, when a matrix has...when one of its 00:11:22.300 --> 00:11:26.800 dimensions is one. So, for example, here you have one row 00:11:26.800 --> 00:11:30.200 and multiple columns. This is actually called a row vector. 00:11:30.200 --> 00:11:32.500 A vector is essentially a one dimensional matrix, where one 00:11:32.500 --> 00:11:35.700 of the dimensions is one. So, this is a row vector and similarly, 00:11:35.700 --> 00:11:38.800 this is a column vector. That's just a little extra terminology 00:11:38.800 --> 00:11:41.400 that you should know. Uhm, if you take linear algebra and calculus 00:11:41.400 --> 00:11:44.200 your professor might use those terms and it's good to be 00:11:44.200 --> 00:11:49.015 familiar with it. Anyway, I'm pushing eleven minutes, so I will continue this in the next video. See you soon.