In 1922, on Gaussian elimination, this is math 1050, College Algebra, I'm Dennis Allison and I teach in the mathematics department here at UVSC. Let's look at the objectives for today's episode. Let's see, first of all, we'll talk about the matrix representation of a system of linear equations and related to that, we'll talk about the three elementary row operations. This leads us to item number two called Gaussian elimination, which is a technique for solving a system of linear equations. And then finally, we'll talk about two special cases, those are the inconsistent systems and the dependent systems. So we'll do all of these things today. First of all, let's go to the green screen and let me remind you how we go about solving a system of equations. Let's take one fairly simple. Suppose we have x plus 2y equals 4 and another equation that says 2x minus 3y equals 1. Okay, now, when we solve a system of linear equations, by the way, this is referred to as a system of linear equations because I have more than one equation in it and these are all linear equations because if I graphed them, they would be graphs of straight lines. And I refer to this as a 2 by 2 system of linear equations. Now when I say 2 by 2, what I mean is that there are two equations and two unknowns. So the first number refers to the number of equations, the second number refers to the number of unknowns. So I've got two equations here and I have two unknowns, so we call it a 2 by 2 system. Now when I'm solving a system of linear equations, there are really only three things I can do to the system that can make it look different but keep the same answer. For example, I can interchange, number one, I can interchange two equations. I can interchange two equations. So in other words, if I just switch equation one and equation two, just invert them there, that wouldn't affect the outcome but it would certainly make the system look a little different. I think we would recognize it but it would still have the same solution. The second operation I can perform is I can multiply any equation by a non-zero constant. For example, if I wanted to eliminate the x's, what I could do is multiply the first equation by negative 2 and I would write it as negative 2x minus 4y equals negative 8. I've multiplied through there by negative 2 and I could rewrite the second equation. Let's see, that's a minus right there. And this looks like a different system of equations. In fact, I guess technically it is but it still has the same solution and the purpose of multiplying through by a negative 2 is so that maybe in this case I could make my x terms add to be to be zero. Okay, so I can multiply any equation by a non-zero constant, by a non-zero constant. So in this case I multiplied by negative 2. The third thing that I can do to a system of equations that won't destroy the solution but will make it look different is I can take a multiple of any equation and add it to another equation. So I can add a multiple of one equation to another equation. Now let me just back up here to item number two. You know in number two it says that we could multiply the equation by a non-zero constant. Why do you think it has to be non-zero? What's wrong with multiplying an equation by zero? Stephen? You get something like 0x minus 0y equals zero and that can be anything. Yeah, for example, you see if I multiply the first equation here, if I multiply both sides by zero I get zero equals zero and what that means is now I pick up solutions here that may not be solutions for both of these because everything solves zero equals zero. I mean this doesn't discriminate, x can be any value, y can be any value but the second equation says the x's and the y's have to satisfy that certain equation. Over here they had to satisfy not only the second equation but the first equation. So when you multiply by zero you essentially wipe out that equation and you've just deleted that information. So you multiply by a non-zero constant so you don't completely wipe it out. And then the third step or the third operation is to take a multiple of any one equation and add it to another. Okay, well let's go to the first example and I'll demonstrate what's called Gaussian elimination using these, oh we have the elementary row operations on a graphic. Let's go to that one. It says, number one I can interchange two rows, we were just discussing that. Number two I can multiply a row by a non-zero constant and number three I can add a multiple of a row to another row. Those are the only three things you can do to a system of equations that won't change the answer to some other answer. So we want to keep these three operations in mind. Okay now let's go to the next example. Here we have an example of a two by two system of equations but I want to demonstrate a new method for solving it that I don't think you've seen before. This is called Gaussian elimination. Now the system of equations says that 2x plus 5y is eleven and 3x minus 2y is seven. Let me write this on the screen here, 2x plus 5y is eleven and 3x minus 2y is seven. Okay so if we come to the green screen, the first thing I'm going to do is I want to abbreviate this system of equations so I don't have to write down so many symbols. You may say but Dennis there aren't that many symbols here to have to write down. Well you know if this got bigger, if this had been like a five by five system of equations, five equations and five unknowns, there'd be a lot of writing involved. So I'd like to abbreviate it as much as possible. So what I'm going to do is put square brackets here and inside I'm going to put these coefficients 2, 5, 11, 3, negative 2 and 7. Now I refer to this as a matrix and not unlike the movie called The Matrix. This matrix is an array of numbers. You'll sometimes see expressed with square brackets. Sometimes you'll see parentheses written on there so you can do it either way with parentheses or with the square brackets. So one thing you wouldn't want to put around it are two vertical bars because that's going to mean something else later on in these episodes. So either parentheses or square brackets. Now I would call this matrix a two by three matrix because it has two rows and it has three columns and the last column represents the constants. In some textbooks you'll see a little dotted line put in right here to separate the constants from the coefficients of the variables. I'll probably not do that because on this green board it just adds more writing and it may make it harder for you to read on the TV screen. But sometimes you'll see a dotted line in there to separate the coefficients of the variables from the constants. So on the first line you can see our first equation. That's 2x plus 5y is 11. We know that the first column are coefficients for x, the second column are the coefficients for y and the last column are the constants. And then the second row says 3x minus 2y equals 7. Okay now with this matrix I want to begin to reduce it so that it eventually has this form. Let me just jump down here further down on the green board and here's what I would like to see. I would like to see a one in this position and a zero below it. And then I would like to see a one in this position and just some number up above it. It doesn't really matter what number goes there. And then on the other side of the dotted line I'm going to get two numbers to appear. Now let me just make up some numbers to fill in these spaces. I don't think they're the actual numbers we're going to get eventually. But suppose there's a 2 here, suppose there's a 5 here and suppose there is a 6 here. Now if I can make this matrix turn into this form then I'll have ones along what I call the main diagonal starting in the upper left hand corner and going down diagonally. Ones along the main diagonal and zeros below the ones. Now if I can get my matrix in this form then what this matrix says as a system of equations is that x plus 2y equals 5. What's just the second equation saying? It says y equals 6. So right away we know the value for y is 6. And then if I back substitute that y into the other equation that says that x plus 12 equals 5 and so x would be negative 7. I don't really think that's the solution for my system of equations but this is just to demonstrate why I want to get my matrix in this form. I can get one of my variables y equals 6 and then I can back substitute into the other equation and I can go back and get x. Now you might say well Dennis how can you make that matrix become this matrix? What I'm going to do is use the three elementary row operations so that I don't destroy the solution along the way. You know there is a term I forgot to mention here this matrix that represents that system of equations. Matrix has a name and it's called the augmented matrix. The augmented matrix that's the 2 by 3 matrix that I've written down here. If I had only listed the coefficients and made a square matrix like this then that's referred to not surprisingly as the coefficient matrix but I've attached the constants as well in the last column. So there's an augmented matrix and there's a coefficient matrix both of these you'll hear me use these terms as these episodes progress. So let's go back to the 2 by 3 augmented matrix for this system. Now what I'd like to do is to get a 1 in this first position and then get a 0 below it and then going to the second column I'd like to get a 1 in this position and then I should have the matrix in the form I'd like. So how can I get a 1 here? Well let's see what are my three elementary operations? I can interchange two rows but I don't see that that would help if I interchange these two rows I'll have a 3 rather than a 2. Another possibility is I can multiply by a non-zero constant and you know I can multiply the first row by a half but I'm going to avoid doing that because that would introduce fractions. If I multiply by a half I'll get 5 halves and 11 halves as well as the 1 in the first position and I'd like to avoid fractions only because it's a little bit more difficult for us to deal with. So I'll stick with integers as long as I can and the last possibility is I can take a multiple of one row and add it to another row. Well here's what I think I'll do I'm going to change this matrix to a new one and I'm going to take a multiple of row 1 and add it to row 2. I'm going to take row 2 and add on negative 1 times row 1. Now you see if I multiply row 1 by negative 1 and add it to row 2 I'll have a 1 in this position and then afterwards I can interchange the two rows and I'll have a 1 up above. Okay so let's do this I'm not actually going to change row 1 I'll go ahead and put in the dotted lines for a while but I am going to change row 2. I'm taking row 2 and adding negative 1 times row 1 that'll give me a 1 here. And if I take this entry and add on negative 1 times 5 I'll get negative 7. And if I take this entry 7 and add on negative 1 times 11 what number will I get there? Negative 4. Okay so what I've done is I've produced a 1 in the second position of the first column. So now I'm going to interchange those two rows. Now the way I'll denote that is I'm just going to put an arrow going from row 1 to row 2 to indicate I'm interchanging those and I'll put 1, negative 7, negative 4 and then 2, 5 and 11. Now at any given moment I could stop, take this matrix, this augmented matrix, write a system of equations for it and solve it but I don't see that solving what I have here would be any simpler than what I had originally. So let's keep going. I have a 1 here I'd like to get a 2 in this position, excuse me a 0 in this position. So what I'll do is I'll take row 2 and I'll add on negative 2 times row 1. Now you see by doing that I'll have a 2 plus a negative 2 is a 0. I'll get a 0 there. I'm not actually going to change row 1 so I'll rewrite what I have. And if I take negative 2 times row 1, add it to row 2 that'll be a 0. That's what I was after. So I have the first column the way I want. Now let's complete this. If I take negative 2 times the first entry and add it to 5 that's going to be 19 I think. And if I take negative 2 times negative 4 and add it to 11 I think that'll be 19 also. You notice I'm not putting equal signs between these matrices.  The reason is these two matrices well none of these matrices are actually equal because for example this matrix has a 3 in the second row first column. This has a 1 in the second row first column. So these matrices have different entries so they're not equal. What they have in common is these represent systems of equations that have the same solution. So I'm not saying they're equal but I'm putting a different symbol in between. Okay now I'd like to get a 1 in this position and I think the simplest way to do it would be to multiply by 1 over 19. So I'll do that next. So this is 1 over 19 times row 2. I'm not going to change row 1 but I will change row 2 and it becomes 0, 1, 1. Well you know this is the form of the matrix I was after. I wanted to get a 1 in the upper left hand corner and a 0 below it. And I wanted to get 1's all the way down the main diagonal. And now I'm going to step out and write the system of equations. The first equation says x minus 7y equals negative 4. The second equation says y equals 1. So there's my solution for y. And if I back substitute into the first equation that says x minus 7 is negative 4. And therefore x is equal to 3. So I could give my solution this way. I could say x equals 3, y equals 1. But if you look at the back of your book, here's what you'll probably see is parentheses 3, 1. In other words, these numbers are entered in an ordered pair in alphabetical order. And this takes up less space in the back of the book than saying x equals 3, y equals 1. Now how shall I interpret that answer? Well this is the only pair of numbers that will satisfy both equations. If I go back up here to the beginning, if I put a 3 in the first position and a 1 in the second position, let me get that a little bit lower so maybe you can see that on the screen, then 2 times 3 is 6, plus 5 times 1 is 5, 6 plus 5 is 11. And in the second equation, 3 times 3 is 9, 9 minus 2 is 7. And that's the only pair of numbers that'll solve that system. Now if I were to graph these lines, and if you might imagine over here if I were to graph these lines, one of the lines, I'll just draw something, might look like that. And the other line might look like that. And where those lines intersect is the ordered pair 3, 1. So this would be 3, y would be 1. So one interpretation is I found a point of intersection of two lines. Another interpretation is I found a common solution for two separate equations. And they all make sense as a response. And this is the answer that you get. Now you might say, well Dennis, actually we've solved things like this before. We didn't use this method. But you might say we've solved these by graphing. We've graphed lines in intermediate algebra, elementary algebra, and found the intersection point. And we've also graphed these by the substitution method and by the elimination method in intermediate algebra. So what's the purpose of going through all this? This seems to take longer than what we've learned before. Well, as these systems of equations get more complicated, let's say you have 10 equations and 10 unknowns, the methods that you've learned in previous courses are more complex and very slow. And this is relatively fast as the systems get more complicated. Let's go to another example and I'll demonstrate this. Let's go to the next graphic. OK, here we have a system of equations. I would call this a 3 by 3 system because I have three equations and I have three unknowns. And this is solved by Gaussian elimination. OK, so I'm going to write the system of equations up here. I'll put it over here in the corner. x minus y plus 3z equals 4. And then there's x plus 2y minus 2z equals 10. And then 3x minus y plus 5z equals 14. OK, this is a 3 by 3 system of equations. One thing I should mention before I begin is why is this called Gaussian elimination? Well, first of all, it's called elimination because this is very similar to the elimination method you learned in intermediate algebra where you add and combine equations to eliminate a variable. When I produce zeros, that's exactly what I'm doing is eliminating a variable. Gaussian elimination because this method was invented by a fellow named Carl Friedrich Gauss a little over 150 years ago. OK, so let's try Gauss's method for this. First, I'm going to write an augmented matrix. So my augmented matrix will be 1, negative 1, 3, 4. And I think I'll put in that dotted line temporarily to separate the constants from the coefficients. 1, 2, negative 2, 10. And then finally, 3, negative 1, 5, 14. OK, so this is the augmented matrix for this 3 by 3 system of equations. And you know that the people who actually use this are people going into computer science. People who are going to be majoring in mathematics and related disciplines. Because whereas the arithmetic in these processes can get more and more involved as the systems get bigger and bigger, if we have a computer do this, the computer can do it very efficiently. Because computers can add, subtract, multiply, and divide very quickly. So it might take just a split second to solve this on a computer. For us, it takes a little bit of time to sort of labor through the arithmetic. OK, so here we go. I'd like to get a 1 in the upper left position. And I have a 1 there. So now my goal is to get a 0 here and here, to get 0s below that 1. After I've taken care of the first column, I want to get a 1 in this position. This is sometimes called the 2, 2 position. It's the second row, second column. So I want to get a 1 in the 2, 2 position. And then I want to get a 0 below it. That would be the 3, 2 position, third row, second column. And after I've taken care of that, I want to get a 1 where the 5 is. That's the 3, 3 position. So I want to get 1s along the main diagonal and 0s below it. And I do this one column at a time. OK, so to get a 0 here and here, I think what I should do is to take row 2 and add on negative 1 times row 1. And then on row 3, I should take row 3 and add on negative 3 times row 1. Because row 1 is OK already. I have a 1 there, so I'll just rewrite that. And if I take negative 1 times row 1, add it to row 2. I'll get a 0 here. That was the whole purpose of this maneuver. Then if I take negative 1 times the second entry and add it to the 2, I'll get a 3. And if I take negative 1 times the 3 and add it to negative 2, I'm going to get a negative 5. And what would the last number be over here? Negative 1 times 4, add it to 10. 6. It'll be a 6, right? OK, now we have the last row to compute. Can anyone tell me what the last row would be? We're taking negative 3 times row 1 and adding it to row 3. Negative 3 times the first row, add it to the third row. What number goes here? 0. 0, right. And what goes here? 2. A 2, right. Now for those people at home, and as well as the students here in the studio, you might imagine this is the place where if there's an error made, it's in simple arithmetic. It's not that the process is complicated, but there are so many additions and subtractions and multiplications, it's easy to kind of miss a number here or there. But of course, accuracy is everything in this process. OK, Susan, you said 0 and 2. What would the next number be? 4. Let's see now, we're multiplying by negative 3 and we're adding it to 5, so I think it'd be a negative 4. See, that's going to be a negative 9 added to 5. And one more number, one more number. What's going to go here? 2. 2, exactly. OK, now you see, if you were to miss any of these arithmetic calculations, you're now solving a different problem. And it's no longer equivalent to the original system. So you may come up with an answer, but it wouldn't be the answer for that system. Now, by the way, when you're doing this at home, don't be surprised if you make a careless error like that, because I make careless errors too and you just try to be as accurate as you can. OK, now continuing, let's see. I have the first column taken care of. I have a 1 at the top and 0s below it. So now I go to the next column and I don't really care what's above the 3, but I do care what's in the 2, 2 position. I'd like for that to be a 1. And I'd like to have a 0 right below it. Now my first goal is to get a 1 in that position. Can anyone tell me a way that you could get a 1 where the 3 is? Matt. Well, you can add to it negative 1 times the third row. Exactly, right. If you take negative 1 times the third row and add it to the second row, that will be a 1. Now an alternative, some of you might be thinking, would be to just multiply the second row by a third. That would give you a 1 and you'd have a 2 over here. But the problem is this would be a negative 5 thirds. And if you think we make mistakes with integers, we really make mistakes when we get to fractions. So I generally avoid fractions if I can. If you decide to multiply row two by a third to get a 1 there, that's perfectly OK. It's perfectly legal and you could do that on your homework or on an exam, but I would recommend doing what Matt just suggested and staying with integers. So let's see, I think I'll come down here to write this. We're going to take row two and add on negative 1 times row three. Now this is my own notation that I've invented for putting out here in front to always remind me how these numbers are being computed. I'm writing this right across from row two. Row one hasn't changed, so I'll just rewrite it. And row three hasn't changed, but row two changes because I'm taking negative 1 times row three, adding it to row two, that's still going to be a zero. And I'm taking negative 1 times 2, adding it to 3, and that'll be a 1. That's what I was after. What will the next number be? Negative 1 times this, add it to negative 5. Close, not 1. It'll be a negative 1. Yeah, because see, multiplying by a negative 1 makes it a plus 4. Plus 4 added to negative 5 is negative 1. And the last entry will be? A 4. Is a 4, exactly. OK, well, we're halfway there. I have the first column taken care of. I have the second column half completed. I need to get a zero right here. How can I get a zero where the 2 is? Take negative 2 times row 2 and add it to row 3. Add it to row 3, exactly. And I'm writing those instructions in front of row 3 because that's the row that's going to change. Row 1 doesn't change. And row 2 doesn't change. I want to keep that 1 where it is. But now I want to get a zero here. You know, I think what this points out is why we want to get the 1 first. Once you have a 1 there, you can use it as a lever to eliminate any other number. So if you want to eliminate a 2, you multiply the 1 by a negative 2 and add it to it. So I got the 1 because that would be easier to work with and say if I had a 3 there, if I had a 3 here and tried to eliminate the 2, I'd have to use fractional multipliers. Well, let's see, this would be a zero and another zero. The next entry will be negative 2. And the next entry will be negative 6. OK, I have column 1 taken care of. I have column 2 taken care of. Now I come to column 3. What would I like to see where the negative 2 is? A 1. Would like to see a 1. And how can you get a 1 there? Divide by negative 2. Right, you can divide by negative 2. I'll say multiply by negative 1 half. You say potato, I say potato. The difference is in our elementary row operations, we said we could multiply by non-zero scalars. I'm sort of thinking of multiplication. But I understand exactly what you're doing. You'd be dividing by negative 2. Let me point out some things that wouldn't work. What if someone said, Dennis, if you take row 1 and add it to row 3, you will get a 1 in that position? That's true, but that's not what we'd want to do. Why not, Matt? Because you'd also have to add the negative 1 to the 0's, and that would eliminate 0's. And then you'd lose those 0's. And we've worked hard to get those 0's. So we don't want to lose the 0's. So I don't think I'd want to add row 1 or row 2, any multiple of it to row 3, because I'll lose the 0's. And you say, well, so why do you want the 0's? Well, just stay tuned here, and you'll see at the end of the problem why those 0's are really very precious. We're going to multiply, better make that a little bit lower. We're going to multiply row 2 by negative 1 half and add it to row to row 3. So it looks like this. My top row is left unchanged, 1, negative 1, 3, 4. My second row is left unchanged, 0, 1, negative 1, and 4. But my last row becomes 0, 0, 1, 3. Well, this is what we're after. We have all three columns in the form we want for Gaussian elimination. Now what do we do with this? Well, now I step out of this augmented matrix and I write the corresponding system of equations. x minus y plus 3z is 4. That's what the first equation says. The second one says that y minus z is 4. And the last one says that z is equal to 3. So now I'm going to back substitute, and I'll get y. Do you see a mistake? Row 2 and row 3 there at the very end. Wouldn't you add something? Oh, yes. I've got the wrong instruction here, thank you. Yeah, I did the right thing. I wrote the wrong instruction. This should be minus 1 half times row 3. That's all there was to it. Yeah, thank you, Jeff. I think I was thinking of so many different additions of rows that I wrote it that way here. We didn't add row 2 to row 3. We just multiplied row 3. OK, thanks. Well, I hope everybody wasn't confused by that. So now I'm going to back substitute. Can you tell me what y would equal here? y minus 3 is 4. What is y going to equal? 7. Y is going to be 7. Let's see, so we have z. We have y. Now I'm going to back substitute in both places up here and go back and get x. I need a little room, so I think I'll erase this last matrix. So in that first equation, we have x minus y plus 3z is 4. And we know that y is 7, so this is x minus 7, plus 3 times 3 is 4. And therefore, x minus 7 plus 9 is 4. So x plus 2 is 4, so x is 2. OK, so you see, at the end of the class, OK, so you see, at this point, things come together very rapidly. And my solution then is 2, 7, 3. I would write that as an ordered triple, 2, 7, 3. And that's the way you'd see the answer probably in the back of the book. If you said, Dennis, I'd like to write the answer. x equals 2, y equals 7, z equals 3. That's OK with me, but I don't think that's how you'll see it in the textbooks, because they try to save a little space when they record these. OK, let's go to the next example. And now we'll see an application of Gaussian elimination to a problem where we have to actually set the problem up in some fashion. This is called a partial fraction problem, and it goes like this. It says write the rational expression 5x plus 7 over x minus 1, x plus 1, x plus 2 in the form of a constant over x minus 1 plus a constant over x plus 1 plus a constant over x plus 2. Now, you might say, well, that actually work. Well, let me give you an example. Suppose I were to take a problem like 1 sixth. If we go to the green screen, 1 sixth. Now, 1 sixth is 1 over 2 times 3. And this is what we refer to as a proper fraction, because the numerator is smaller than the denominator. Now, whenever you have a proper fraction and you factor the denominator into two factors, like 2 and 3, you can write this as some constant over 2 plus some constant over 3. And these will both be proper fractions as well. If that's a proper fraction, this will be a proper fraction. Now, in this particular case, I happen to know what the answer is. It's not too complicated. I think we should put a 1 there and put a negative 1 over here.  Let's try this one more time. Suppose we had 5 over 21. Now, 21 is 7 times 3. I'll put the 3 first. 3 times 7. I'll put the smaller than the larger. Not that it really matters. And so a proper fraction with a denominator that factors into two different factors, like a 3 and a 7, can be written as something over 3 plus something over 7. And that sum, I bet one of the numerators could possibly be negative, should equal 5 over 21. Now, what would that be? Well, let's see. Let me just play around with some things here. What if I put a 1 here, 1 third? How many 21sts is 1 third? 7. That's 7. OK, 7. So what I need to do is subtract off 2 21sts. I don't see how I can make that 2 21st. So let's put a 2 3rds here. Put a 2 3rds here. How many 21sts is 2 3rds? 14. 14 21sts.  Now you know 9 over 21 is the same thing as 3 over 7.  I think that should be a negative 3. Now you notice this is a proper fraction and this is a proper fraction. And that sum is equal to 5 over 21. That's not a fact that you necessarily hear about in previous courses, but it's a fact that you will use later on if you take a course in calculus. Let's go back to our example and see how we're going to use this fact to solve this partial fraction problem. We have 5, let's see, 5x plus 7 over the product x minus 1, x plus 1, and x plus 2. And this is a proper fraction in the sense that I have the smaller degree on top. So I should be able to write this as a sum of proper fractions, something over x minus 1, something over x plus 1, and something over x plus 2. And these are proper fractions as well. That means the numerator has a degree smaller than the denominator. But these denominators are first degree. So the numerators must be 0 degrees, in other words, constants. So I'll just say a, b, and c for those constants. So the question is, what should a, b, and c be in order to decompose this original fraction? These are called the partial fractions because they each play a part in composing the original fraction. Well, to figure out what a, b, and c are, I'm going to multiply both sides by the product x minus 1, x plus 1, and x plus 2. Now, if I multiply on the left by that, I get 5x plus 7. And if I multiply on the right by that product, well, different things happen with each expression.  And then in the second fraction, if I multiply by the three factors, I get b times x minus 1 times x plus 2. And in the third situation, x plus 2 cancels, and what would be left? C times x minus 1 times x plus 1. Times x plus 1. OK, now I'm going to multiply out the left hand, the right hand side, and collect terms.  OK, so that's going to be a x squared plus 3ax plus 2a. This is just a bunch of basic algebra here. I'll multiply by b, and I get bx squared plus bx minus 2b. I know you've been wondering 2b or not 2b. Negative 2b is the answer to that. OK, so this is going to be cx squared minus c. Now, collecting terms, how many x squares will there be? How many x squares are there? Well, I've got ax squared plus bx squared plus cx squared. That's a plus b plus cx squared. How many x's are there? Well, I've got 3a plus b, and that's it, 3a plus b. And then finally, there's a constant term.  Now, you remember on the left-hand side, this is equal to 5x plus 7. So here's what I can conclude. The constant term here has to equal 7. The constant terms have to be the same. What do you think 3a plus b is going to have to equal? Has to equal 5, because this is how many x's, and I have 5x's.  Zero. It's got to be zero, because there aren't any x squares over here. So a plus b plus c is zero. 3a plus b is 5. 2a minus 3b minus c is 7. I want to solve that system of equations, and then I'll know what those numerators should be. Let me just write down that information here before I lose it in this space. So what we have is that a plus b plus c is zero.  And we have 2a minus 2b minus c equals 7. OK, I want to solve this using a system of equations. Now, I think I'll stop here and go over to the board, because I'm going to need more room to do this. So I'm going to walk over there to write this down. OK, so over here at the board, I've listed our system of equations. It's a 3 by 3 system. And I'm immediately going to change it to its augmented matrix. So the matrix is 1, 1, 1, 0. And then 3, 1, 0, 5. And then 2, negative 2, negative 1, and 7. OK, so watch me carefully, because it's easy to make careless errors on these things. Looks like I have a 1 OK, but let's get zeros where the 3 and the 2 are. I can do that simultaneously.  I think that'll give me zeros here and here. The first row doesn't change. It's 1, 1, 1, 0. But the next row will be 0 and negative 2 and negative 3 and 5. The 5 doesn't change because we're adding a multiple of 0 to it. And on the bottom row, I'm going to get 0 and negative 4 and negative 3 and 7. OK, so the first column is taken care of. Now in the second column, I'd like to get a 1 there. The first thing that comes to mind is I could multiply by negative 1 half. But if I do that, I'm going to get fractions over here. And I'd rather avoid introducing fractions as long as possible, because it's easier to make careless errors with those. So I think I'll do this. What if I were to subtract? Well, what if I were to add negative 1 times rho 3, add that to rho 2? And I think all these odd numbers will become even over here. So let's try that. I'm going to take rho 2 and add on negative 1 times rho 3. Now, rho 1 doesn't change. And rho 3 doesn't change, in this case. But rho 2 does. I'm adding negative 1 times rho 3 to rho 2. Well, negative 1 times 0 plus 0 is still 0. So my 0s haven't changed. But the negative of negative 4 added to negative 2 is plus 2. And I think the rest of the rho is going to be a 0 and a negative 2. Well, you see now, all of these numbers are even. So I can now multiply by 1 half. And I'll still have integers. So this will be, let's see, I'm going to take 1 half of rho 2. So that's going to be 1, 1, 1, 0. 0, 1, 0, negative 1. And the bottom row is 0, negative 4, negative 3, 7. OK. Next, I've got a 1 there. I need to get a 0 right below it. So I'll add 4 times rho 2 to rho 3. Let's do that back over here. Rho 3 plus 4 times rho 2. The top row hasn't changed. The second row hasn't changed, but the bottom row does. 0, 0, negative 3, and 3. Well, you know, now the only step left to do is to get a 1 in this position. So what I'll do is take negative 1 third of rho 3. And my matrix is 1, 1, 1, 0. 0, 1, 0, negative 1. And 0, 0, 1, negative 1. OK. This is the augmented matrix that I wanted to arrive at. And I'll now solve that to get the A, B, and C. Let's go back over to the green screen. And we'll finish this up. And we can substitute it into our fractions as well. OK. I'm going to write down that augmented matrix. I'll just copy it over here. It's 1, 1, 1, 0, 0, 1, 0, negative 1. And then 0, 0, 1, negative 1. OK. So I'm ready to step out and write down the new system of equations. That would be A plus B plus C is 0. And then there's B equals negative 1. And there's C equals negative 1. Well, we're in luck here because there's no back substitution to make into the second equation. I have B and C completely isolated. So I'll substitute them both in the first equation. A plus negative 1 plus negative 1 is 0. So A is 2. So what that tells me is the numerator of the first fraction should be 2. The numerator of the second fraction should be negative 1. And also the third fraction negative 1. So if you take 2 over x minus 1 minus 1 over x plus 1 minus 1 over x plus 2, that is a partial fraction decomposition of the original fraction given back here. Now, this is a technique that's important if you take calculus. I know not all of you are going to take calculus. But on the other hand, the process of reducing matrices using Gaussian elimination is important to students who are going to major in business, in computer science in particular, and in other related areas in the sciences. Knowing about a matrix reduction is important there. OK, let's go to another example. And in this example, I want to illustrate that a system doesn't have to be square. I've done 2 by 2s and 3 by 3s. Here's a system that's a 2 by 4. And I want to show you how to solve this. And I think we can do this one on the green board. One of the things to point out, though, is that when you have a system that has more unknowns than you have equations. So we have 4 unknowns and only 2 equations. What that means is this system cannot have a unique solution like x equals 2, y equals 3, z equals 0, w equals 4. It's going to have either no solution or it's going to have infinitely many solutions. And let me show you how that will play out in this problem. Now the system of equations says x plus y minus 4z minus w equals 5. And the next equation says x minus y plus 2z plus w equals 1. Now when I write my augmented matrix, it's going to look a little different from the others. It's going to be 1, 1, negative 4, negative 1, 5. I better put that dotted line in there. It's getting so long. And the other equation is 1, negative 1, 2. That's a z there. Plus, well 1, don't need to put a plus on it. And a 1. OK, now the strategy is still the same. In the first column, I get a 1 at the top, which I have. And I get a 0 below it. And then in the next column, I get a 1 in the 2, 2 position. And I don't really care what's above it. And that's where the problem ends. These other columns I'll not be able to complete because they don't come further down. My main diagonal does not extend. So this will be a relatively short problem. Let's begin by putting a 0 in the 2, 1 position, second row, first column. So I'm going to take row 2 and add on negative 1 times row 1. And here's the matrix we get. 1, 1, negative 4, negative 1, that's a 5. And here I get a 0. And I get a negative 2. And I get a 6. And I get a 2. And I get a negative 4. I want to get a 1 right here. Well, all these numbers are even. So multiplying by negative 1 half is no problem at all. Let's do that on the next line. So this says, if I take negative 1 half of row 2, this will look like, on the second row, 0, 1, negative 3, negative 1, 2. OK, but what does this mean? Well, if I step out and write my new system of equations, I have x plus y minus 4z minus w is 5. It turns out that equation never changed. That's exactly the original equation I had up here. The other equation says y minus 3z minus w equals 2. That's a 2 there. OK, what can we make of this? Well, let me just show you. I'm going to erase the rest of the screen here. And let me put that new system of equations over here. x plus y minus 4z minus w equals 5. And then y minus 3z minus w equals 2. OK, so how are we supposed to interpret this? Well, here's what we do. In the second equation, you solve for y. y is equal to 3z plus w plus 2. Yeah, I've got to add 3z to both sides. I've got to add w to both sides. I'm going to back substitute into the other equation. And that says x plus y, and I'll put this in for y, 3z plus w plus 2.  Now, if I collect terms, this says x minus z. The w is canceled out, plus 2 equals 5. OK, in this equation, I'm going to solve for x. x is equal to z plus 3. OK, so I got this equation, and I got this equation. And somehow out of this, I'm supposed to establish an answer. Well, my answer is as follows. x equals y equals z equals w equals. Those are the four variables in the problem. What does x equal? Well, x is equal to this. Let's see, z plus 3. What does y equal? y is equal to this. 3z plus w plus 2. What does z equal? Well, for lack of a better name, I'll say z equals z. And I'll say w equals w. So how do we interpret all that? Well, you notice on the right-hand side, I have only z's and w's. Those are called parameters in this answer. z and w are called parameters. And what this says is you can pick any value you want for z and for w, and plug them in here, and calculate the corresponding x, y, z, and w. And you will always get answers to that problem. For example, suppose we let z equals 0, and suppose we let w equals 0. What would be the corresponding x, y, z, and w? Well, if z is 0, what is x? 3. 3. And if z and w are both 0, what is y? 2. 2. And then we said z would be 0 and w would be 0. This is a solution to the problem. But it's only one of many. You could pick other answers this way. Suppose we decided to let z be 1 and let w be 2. What would be the corresponding x, y, z, and w? Well, if z is 1, x would be 4. And if z is 1 and w is 2, I think that turns out to be y equals 7. And then we said z would be 1, y would be 2. This is another solution for the problem. So you see, we could keep doing this indefinitely. We could just keep picking new z's and new w's and just coming up with new answers. So how are we going to record this answer, say if you were to look in the back of the book? The answer would be recorded this way. I'll put in an x, a y, a z, and a w as an ordered four-coordinate point. I'd put z plus 3. I'd put 3z plus w plus 2. I'd put z and I'd put w. And that tells me this is the x, this is the y, this is the z, this is the w. And there's no way I can list all the solutions, except to give them in this form. Now, when you have more equations, when you have fewer equations than unknowns, then you're going to come up with a system of equations that looks like this. I'll be doing another one of these in the next episode. So this is just by way of introduction. How would I know if there's no solution for the system? Well, let me just back up and show you what would happen if there were no solution. Suppose you're in the middle of reducing a matrix and you have, let's say, a 1 here. I'll just fill in some other numbers, 2, negative 5, 3. And you have 0s below this. Suppose in the next equation, you have a 1. And then you have a 4. And over here, you have a negative 2. And suppose in the last, oh, you get a 0 below that. Now, our goal would then be to get a 1 in this position. But suppose you get a 0 there. And over here, you get, let's say, 6. Well, I don't see any way I can get a 1 there. I can't bring in multiples of other rows here, because I'll lose my 0s. In fact, what does this equation say right here? 0 equals 6. It says 0 equals 6, which, of course, is a contradiction. So if your matrix reduces to something that has a contradiction, then you can stop right there and just say, there's no solution. Those are the shortest problems of all if you see a contradiction. But you know, another possibility is what if there is a 0 here as well as over there? In this case, the equation says 0 equals 0. And that's certainly true. That's not a contradiction. What that means is you can eliminate that row, because there's no new information in it.  Now, because I wasn't able to get a 1 in the next position, that tells me this is going to have infinitely many solutions. I'll have to use a parameter in my final answer and get x and y in terms of z. OK, well, we're sort of in the middle now of a big question about how to solve the system of equations using matrices. We've looked at a method today called Gaussian Elimination. Gaussian Elimination produces 1s along the main diagonal and 0s below it. And then you use back substitution to solve for all the other variables that you don't get immediately at the bottom. But this can sometimes lead to problems with infinitely many solutions. And we'll look at some more of these sort of cases next time. And you can occasionally get a problem with no solution. And so when we meet in the next episode, we'll talk about an alternative to Gaussian Elimination that takes a little bit longer, but it doesn't require the back substitution. So in one sense, it's longer. In one sense, it's shorter. And that's called Gauss-Jordan Elimination. And we'll look at all of these cases again and see what happens in those situations. So I'll see you for the next episode.


from Las Vegas, it's theCUBE. Covering EMC World 2016. Brought to you by EMC. Now, here are your hosts, John Furrier and Dave Vellante. Okay, welcome back everyone. We are here live in Las Vegas for EMC World 2016. This is SiliconANGLE Media's theCUBE, our flagship program. We go out to the events and extract the signal from the noise. I'm John Furrier, my co-host Dave Vellante. It's our seventh year, theCUBE started in 2010, and we interviewed our next guest during that time. I think every year since, Howard Elias, welcome back to theCUBE, President and COO of Global Enterprise Services for Dell EMC, and also leading, co-leading the integration between the merger. Howard, great to see you. Congratulations to you, John. Thanks, appreciate being back. It's a big job. It's kind of like a big construction project. You're like the foreman on the job. You got to make sure everything runs on time, effectively. I mean, the biggest acquisition in tech history. It is. We're going down a lot of conversations. I mean, there's whatever it's talking about.  What's the status? Is it going to close on time? What's the stock price going to be? Oh, it's not top of the top. We'll talk about a lot of things, probably not that. We're going to get edited out by the, seriously, on a serious note. Big project. It is. Two cultures aligned. What's your thoughts on what people are talking about and what would you like to clarify? Well, let's start at the top. As you said, it's the largest tech combination in history. It'll result in the largest privately controlled integrated IT company on the planet. Pretty exciting stuff. We're part of making history here. And it's actually great to see the integration teams have come together very well. And at the highest level, the transaction is on plan, on track, on the original terms, on the original timeline. So when we said back in October, the expected close, we said it would be sometime in Dell's Q2 or Q3. So between May and October, we're right on track. The vast majority of the regulatory approvals have occurred. In fact, there's only a couple of things left to have happen. The SEC will finalize their review of the S4 registration statement. This is the statement that Dell will use to issue the tracking stock. Michael Meadows of International Things. China, right? Well, I was going to get to that, right? So the SEC, just to get the registration statement finalized in the final proxy statement, that needs to happen, and we're very close to that. Then comes the shareholder vote, where the EMC shareholders actually get to vote on it. And then the last of the regulatory approvals is China. We have the vast majority of all the other ones. But everything's tracking exactly as we expected. And then once that happens, the teams will set a closing date. So EU's cool. EU's already done. US is already done. All the other major countries around the world that typically weigh in on something like this. And it all happened to completely on track. How would you compare and contrast this transaction with others you've seen in the industry? Because a lot of people like to, I mean, first of all, people will have opinions on M&A organic growth versus inorganic. This is a big monster transaction, as you pointed out. What is this like, if anything, like anything others? Or how isn't it like other big M&A deals? Once certainly you've been involved. It's a great question, and you may know I was actually involved in the DEC compact combination as well as compact and HP. So I have a lot of experience and battle scars along the way. And look, here's what makes this one very different. First of all, it's larger, but it doesn't actually mean it's more complex. The complimentary nature of the product technology portfolio. Very complimentary, extremely little overlap. There is some overlap in the mid-range and low-end of storage. But if you think about client and server and storage and CI and virtualization and cloud infrastructure and cloud-native apps, no overlap. So that, you know, it bodes very well, right? Number two, very complimentary customer segments and go to market. While we all sell in all markets, EMC is really strong in the large enterprise, large public sector, large telco, large service provider. Dell's strength really lies in mid-market and SMB. You know, third, the cultures are very, very similar. You heard Michael talk about that at his keynote this morning. And I've seen the microcosm. We're bringing 170,000 people together. Well, we've had several hundred people together now for six months, working across 16 teams, building the integration plans, getting ready for day one and beyond when we do full integration. And these teams have really come together as one. Very open, constructive, collaborative. We focus on customers. We have a desire to win. We're results-oriented and we act with integrity. And then finally, let's not forget, EMC and Dell had a long-term partnership, nine years, that at its peak, was worth a couple of billion dollars. Michael and Joe know each other very well. The management teams know each other very well. And we've demonstrated how we deliver value to customers. How about the IT piece? It was also announced that you will, after the merger, run IT for the merge company. Huge, you know, task. So congratulations, I think. So I want to ask you about, you just saw HP split apart, right? And actually the IT piece seemed, anyway, from an outsider to go pretty well. I remember, you guys might remember this, when sun transitioned from its Amdahl mainframe to its Unix sun systems, it was like a disaster for three quarters in a row. Is IT just better today? Are there better processes? Is it just more mature? And what have you learned from observing some of these other, either mega mergers, I've talked to once before, but even HP split, is IT just better? Do we know more about it now? Yeah, first of all, I will want to say, until the transaction closes, I'm still have my day job here at EMC and co-leading the integration. So yes, post close. Michael has asked me to look after the overall global services and IT. And look, as we all know in talking with customers, not all IT is created equally. Depends where you've started, what, where you are in the journey. The good news, and I've worked with the two teams coming together here, they're of like minds. But the very interesting thing is, just like the complementary nature of the overall strategic rationale coming together, same thing is true for IT. EMC's IT is really built on a lower volume of very high value, complex configurations, integrated solutions, converged infrastructure to deliver. The Dell IT was really purpose built for very high volume, extremely large number of transactions on a daily, weekly, monthly basis. A lot of attach going on with that. And so we're going to preserve and protect the value that each of us have created, even as we look for ways to bring together common customer, master, common reporting. We're going to be very thoughtful in the approach. And the good news is, the IT being agile, flexible, modular, that will allow us to do that. So they always say people process technology. Technology is the easy part, it's the people and process. It's the people and process. Is that what you expect to spend your time? It's exactly what we're spending 100% of our time, people process and the operating model. Because the operating model, how you interface with the product teams, how you interface with the business, how you interface with different customers and different customers' segments. Everybody looks at their, what does it mean to be a trusted advisor to me? Well, if you're a large enterprise or a mid-market customer, it could mean different things. John and I have been talking a lot about the operating model, it's cloud. Cloud isn't a place to put stuff. It's not a place to put data or apps. It's the operating model. It's the style of computing. So how will cloud play into that operating model? The same way we work with our customers. It'll be a hybrid cloud environment. We'll have private cloud instances for agile, flexible, where the workloads are predictable and especially for those mission critical applications. We've already started to introduce VirtuStream even on the EMC IT side to be able to make the choice whether it's our data center or a different data center. But we'll also utilize SaaS-based offerings, partner-based offerings, and even public clouds for the workloads that make sense. Howard, I got to ask you because a lot of people talk about M&A and deals like this, like changing an airplane engine out at 35,000 feet. It's very difficult. And so there's a non-disruptive, operational kind of mindset or guiding principle. So my first question is, two-part question, one is what is the guiding principle of the co-team that you're co-leading with your Dell cohort on this? What's the kind of the guiding principle? And what is the hardest areas to work on? Is it sales teams? Is it operations? Is it the product? Is it IT? How do you stack ranked the challenges? It's a great set of questions. Let me start by, we actually did, from day one, set out those guiding principles. First, it is do no harm. Both companies are great companies in their own right and have created tremendous value for the customers they serve. So protect the value that we both create, Dell with the Dell products and the Dell go-to-market, EMC, same thing. Second, let's make sure we deliver a phenomenal employee experience. The most important thing we can do on day one is have employees feel like they're part of a new company. It's not part of the Dell part of the company or the EMC part of the company. Unify everybody. It's get to everybody to a common vision, a common mission. We understand what we're trying to accomplish. But even simple things like, let's make sure on day one, that employee experience of collaborating, being able to message each other, be able to work together with the tooling is all very important because if you have great employee experience, then you can focus on the next big thing which is the customer experience. So we want to make sure the customer is gets everything they get today, but then the other thing that we'll do very close to day one is show the power of the combination through new offers. New product offers, new solution offers, new service capability that we're going to be able to deliver. So that's really the focus we've had the guiding principle. What are the watch out points? We have got 16 work streams working on go-to-market services, IT, supply chain, operations, you list them all. The most important thing that Rory read, my colleague and I are focused on is majoring on the majors. Make sure we all understand this is about building that great company. And it's really the five things that we're focused on. It's that employee experience. It's the customer and the partner experience. It's the new capabilities that we're going to show the power of this combination. And then it's the revenue and the cost synergies. Now, every big combination like this is going to have some cost synergies. There's going to be certain areas of duplication and redundancy. But we believe the revenue synergies are going to be at least three times the cost synergies. And frankly, the more time we spent working on this, the more opportunity we see. So it's- Everyone's smiling. Well, but it should be no surprise. If you go back to the thesis, complimentary products, complimentary go-to-market. Well, that's a question that's come up in our community. It's a security question, not a technology question. It's sort of a business question. You'll be, I presume the CIO, you're not the CIO. The CIOs will be reporting into you. Yes. A conversation in our community is, how is the conversation changing between the CIO or CISO and the board? What should the board be know about security? How should the CIO, CISOs be communicating to the board? So if you have to communicate to the board the new merged company about security, what should they know? It's a great question. I actually do sit on an outside public board as well. So I have this experience from the other side. And to a board member, it's really part of the risk management framework, right? You have to look at things like reputational risk, operational risk, market-based risk, financial risk, and now cyber risk. So it's part of the overall framework the board has to look after. And both the CIO and the CISO has to be in a position to be able to clearly and articulately explain to the customer, here is the risk profile that we have and here's what we're doing to mitigate the risk. You can't eliminate all risk or otherwise you won't do business. But it's understanding the B2B, the B2C parts of your business. It's understanding those attack surfaces and rolling it up into an overall enterprise risk management framework and get folks comfortable about where you're going to take the risk and where you're not going to take the risk. Is there an increasing recognition that it's not a matter of, if we get compromised but when and then how should we be responding to that? Is that a shift that's occurring? That shift has happened years ago. There's only two kinds of customers left in the world. Those that know they've been compromised and what they're doing about it and those that don't yet believe or know that they've been compromised. The bad guys are everywhere and by the way there's some bad guys inside our companies, right? And so it's how do you actually manage the risk profile? How do you either think about segmenting your networks, classifying your data but at the same time we've all got to do business in an open collaborative way. I want to ask you another question. It's VCE related, Acadia related. You started off- Oh my goodness, way back. A lot of people don't know that but you incubated, you were the original team. You led that team to create that. As you look back, how does it feel? What are your thoughts on what VCE has become? The whole converged infrastructure industry which you guys certainly started or one of the few that were involved early on. What are your thoughts on that? Oh thank you for that. It feels awesome. It's great to see, you know, we started it. My colleague and I, Gary Moore at Cisco and Rob Lloyd and a few of us got together and had this great idea. And look, I don't know that we would have described it the way we do today back then but it actually is the best way of putting a point on what we had in mind and that is why would a customer buy, you know, build what they can buy? And so the fact that we're able to integrate a more complete experience around the infrastructure and the data center where customers can focus at the app level on the people process and operating model and frankly on the digital transformation journey that every customer has to go on. And it's great to see the multiple billions of dollars of results, customers voting with their dollars each and every day. But it's also great to see the expansion of the product line, the expansion of the team, blocks, racks, appliances for more traditional systems of record apps but also now for systems of engagement and system of insight apps. The new VxRail is just a home run and VxRack for those scale out applications as well. It's just great and the team has also matured. I know you're going to have Chad on at some point here. He'll do a much better job of explaining the future excitement going on there. Well, you guys made a good call there because now the Rage is purpose built in our engineered type system. It's an Oracle leading with that and now you're seeing some of the innovations is not about the commodity hardware anymore. No, it's really about delivering that engineered system and customers I think are finally over the fact that they don't have to tweak every knob and dial that they just want to now buy and implement versus build and focus their dollars and their resources elsewhere. Howard, thanks so much for coming on theCUBE. I know you're super busy at the integration. I'll give you the final word. What is the, for customers watching and certainly employees of both companies, give a quick sound bite on the vibe of the integration, the plans, et cetera and share some color on that. Well, you know, it's great. I'm out with customers all the time. There's been research that have been done as well. The vast, vast majority of customers are looking forward to the combination. We're humbled and pleased to know that most of those customers actually say to us that they believe they will spend more money with the new combined company than they spend individually. And I think it's because they see that value as well. As customers go on their digital journey, as they modernize IT, as they think about the future trends of converged infrastructure, software defined everything, cloud infrastructure, cloud native, next generation apps, all done in a secure and mobile way, this will be the essential infrastructure company for this new industrial revolution. There will be no better company on the planet that can deliver that integrated solution to customers. Howard Elias been there, done that and now driving the bus on the integration, man on the job, big task, congratulations. Good luck, continued momentum. Hope we get closed on time. It looks like it's going to. Thanks for sharing your insights here on theCUBE. Live here at EMC World, I'm John Furrier with Dave Vellante, extracting the signal from the noise. You're watching theCUBE. It's always fun to come back to theCUBE.


Hello and welcome to the session. In this session we are going to discuss matrix representation of a vector, its components and multiplication of vector by a matrix. Now first of all we'll discuss vectors on coordinate plane. We know that a vector is a quantity which has both magnitude or length and direction. It is represented as directed segments such as vector AB. A vector on the coordinate plane can be described in component form or in terms of horizontal and vertical components where horizontal component is the x coordinate or the x component and the vertical component is the y coordinate or the y component. If we are given vector AB as 2 3 that is the ordered pair 2 3 on a coordinate plane then we'll represent vector AB as directed line which starts from the region that is the ordered pair 0 0 and ends at the ordered pair 2 3. We say that vector 2 3 has initial point as the ordered pair 0 0 and terminal point as the ordered pair 2 3. Now let us discuss matrix representation of vectors. A vector can be represented either in column matrix or in row matrix. So we can say that a matrix with dimension 1 cross 2 or 2 cross 1 can be used to represent a vector. So vector 2 3 can be written in column matrix form as a 2 by 1 matrix containing elements 2 3. Similarly vector 2 3 can be written in row matrix form as the 1 by 2 matrix containing elements 2 3. Mostly we write it in column matrix form. So vector 2 3 is written as a 2 by 1 matrix containing elements 2 3 with 2 is the x component and 3 represents y component. And now we'll see how to determine component form of a vector if vector A is given in the form of 2 by 1 matrix containing elements minus 3 2 and vector B is given as a 2 by 1 matrix containing elements 5 4. Then we need to express vector A B in component form and also we shall graph vector A B. Now x and y components of vector A B are found by finding B minus A. So B minus A is equal to B that is the 2 by 1 matrix containing elements 5 4 minus A that is a 2 by 1 matrix containing elements minus 3 2 and this is equal to a 2 by 1 matrix containing elements 5 minus of minus 3 4 minus 2 which is equal to a 2 by 1 matrix containing elements 5 minus of minus 3 is 5 plus 3 that is 8 4 minus 2 is 2. So B minus A is equal to a 2 by 1 matrix containing elements 8 2 or we can write it as vector A B which is given by B minus A is equal to a 2 by 1 matrix containing elements 8 2 where 8 is the x component and 2 is the y component. So its component vector form is the ordered pair 8 2. So we graph vector A B which is given by the ordered pair 8 2 it will start from the initial point that is the ordered pair 0 0 and will be extended to the point with the coordinates 8 2 which is the terminal point and we show the direction sign by this arrow. Now we shall discuss how to multiply a vector with a matrix. We have discussed vectors on coordinate plane having 2 components. Now vectors can also be 3 dimensional that is having 3 components and its matrix will have dimension 3 cross 1 or 1 cross 3. We know that we can multiply 2 matrices A and B if and only if the number of columns of matrix A is equal to number of rows of matrix B. Now suppose we have matrix A of order 2 cross 3 and we want to multiply it with a vector then the vector must have 3 rows in its column representation. It means vector will be 3 dimensional that is it will have 3 components. Now let us take an example. Suppose we have matrix A which is given by a 2 by 3 matrix containing elements in the first row as 1 2 3 and elements in the second row as 4 5 6 and if we want to find the product A B where B is a vector and number of rows in vector B is 3 and the order of this vector matrix is 3 cross 1. We know that we can multiply 2 matrices A and B if and only if the number of columns of matrix A is equal to number of rows of matrix B. Here we need to find the product A B and order of matrix A is 2 cross 3 and order of matrix B is 3 cross 1 and here we should note that number of columns of matrix A is equal to number of rows of matrix B. So we say that the product A B is defined and our resultant matrix will be of order 2 cross 1. So here we have a 2 by 1 matrix containing element in the first row as 2 obtain element in row 1 in the resultant matrix. We multiply elements of row 1 in matrix A with the corresponding elements of this column matrix that is matrix B and we have 1 into 1 plus 2 into 0 plus 3 into 1 to obtain element in the second row of the resultant matrix. We multiply elements in row 2 of matrix A with the corresponding elements of column 1 in matrix B and we will add the product and therefore we get 4 into 1 plus 5 into 0 plus 6 into 1 and this is equal to a 2 by 1 matrix containing element in the first row as 1 into 1 that is 1 plus 2 into 0 is 0 plus 3 into 1 is 3 and element in the second row as 4 into 1 is 4 plus 5 into 0 is 0 plus 6 into 1 is 6 and this is equal to a 2 by 1 matrix containing elements 1 plus 0 plus 3 that is 4, 4 plus 0 plus 6 that is 10. So product A B will be equal to a 2 by 1 matrix containing elements as 4, 10 so we get a new vector that is the ordered pair 4, 10 having 2 components. Thus in this session we have learnt matrix representation of a vector, its components and multiplication of vector by a matrix. This completes our session. Hope you enjoyed this session.


What's up everybody? Jason here for jasperjasecasebreaks.com. 2020 community select football, 3 box, hobby, break, picker teams number 18 just sold out. Remember all card shift guys, XRCs are randomized to entire break. The more teams you buy then guys, the more chance you have to win. Remember Trevor Lawrence is an XRC and he will definitely be money. Money signs. 17, 18, 19, and 20 are from the same case. So this is break number 18 guys. If you look on the website number 19 should be posted up and and all 32 teams are still in there as well. Check our break schedule Carlos. It's always dropping every five minutes and if it is sold out, you'll see it on there. And here you go. Three box break guys. Here's customer names. Kevin got the stealers last time on Joe. This lighting is amazing. My's were almost like not as adjusted yesterday. I see a green. I think greens are out of five. All right guys, good luck. We got Henry Ruggs to start us off. Andrew Thomas and a Jalen Hertz color That is uh, red looks more like verminy, but red 127 out of 149. Kalio Max Silver. We got Ryan Tannehill and T Higgins. Five out of five patch. Very nice. Cincinnati Bangos went to Sean. We have Darrington Evans as a field level silver. Little Joey B. Joe Burrow. Tonio Gibson. Wow CD lamb four out of five. What a box. Four out of five green for the Cowboys and that is going to SKS and look at that right behind it. Of course, when you have a cowboy, you've got to have an eagle, right? Jalen Hertz little patch autograph. What a box. 126 out of 189. This box is starting off hot. These are technically the first four hits. Jalen Hertz out of five T Higgins. CD lamb out of five. Jalen Hertz. Michael Piven Jr. CD lamb. Devin Juvenay blue. Cortland Sutton. Ryan and IUke field level. We got Damien Arnett. Silver. We got Bechton with a tri-color to 189 and Kolk met to 35. That's really nice as well. Terry Bradshaw. Wow, is this another green? No way. Kyler Murray. One out of five jersey number one out of five. Wow, this is one of the most loaded boxes. That is a green phenomenon. One out of five jersey number for the Cardinals going to Alex. Kyler Zeehler. That is a premiere level silver as well. LaVisca. Chanel Jr. Michael Piven Jr. We got Brandon IUke. Silver. That is a club level and a premier level, or sorry club level and a concourse as you say. Henry Ruggs. Silver rookie selections. Antonio Gibson club level silver. Justin Jeffers in there. James Connor. 1099. And a silver. Jalen Hertz. Which is a concourse. Justin Jefferson. Jaden Young Clowney. Drew Brees. Trayvon Diggs. And Kolk met. Ah, what a box. All one box guys. I can't. This is wild. Quick recap. The Jalen Hertz is numbered to 189. You got a T Higgins five patch. CD Lam out of five. Jalen Hertz color. One out of five. Kolk met at 35. Petro Zehler. Silver Jalen Hertz as well. What a great box. Next one. Can I even top the next box? Honestly guys, I did a lot of this select as personals last week. And there was one box guys. I hit a green Jalen Hertz. Out of five. Field level. And a Zebra Patrick Mahomes in the same box. Like that one was probably one of the most wildest boxes I've seen. But this one got you three greens. Three greens. And the patch autograph was Jalen Hertz as well. Of course I hit the out of five Justin Herbert black gold autograph. Which is probably one of the biggest monsters of them all. I hit a couple snake skins. But I thought I'd be able to hit a one on one though. I hit a Rob Grandkowski snake skin two out of two. Josh Jacobs is 75. We got Michael Gallant. Michael Pimmin Jr. Silver. Patty Mahomes. CD Lam. CD Lam field level silver. We got TJ Watt. That is a red. Out of 99. And Drew Grease. 35 out of 175. And we got Chase Claypool to 70. Actually out of 25. That is a tie dye three color patch last spot mojo. Seven out of 25. Look at that. Where's Kevin at? Steelers last spot mojo. Look at that. Beautiful. Tie dye. KJ Hamler. Not in the NFL. Any more probably. Calvary's ruined his career. I never actually had the time to go look it up. I used to say the Nucky. But then someone said no. So I'm just like whatever. Damn Marina. But you guys just ruined his confidence. You know what? He was throwing in an emergency situation. Both top two quarterfights going out. Throw him in there. He just gets destroyed out there.  We're not even going to give you another chance bro. Just practice what guy come up. That's literally what they did to the guy. It was so horrible. Speaking of one of one. I said I haven't hit yet. Zach Moss one of one. For the bills. Colin Adams with the bills. Wow I actually just called it. I said I haven't hit a one on one. I'd love to hit one.  Kevin DuVernay. John Taylor. Kyla Murray Phenomenon. We pulled a green earlier. Jerry Judy that is a premier level silver. Exactly dude. How massive is that? So horrible. Baker Mayfield out of 149. Devin Asiasi. J.K. Dobbins. Silver. Joe Burrow rookie selections. Jerry Rice. Daniel Jones little club level. KJ Hamler. Joe Burrow. I think we went two for two on these boxes. Can we go three for three? Alright last box guys. Remember I did repost up the next select. Which already have a couple spots taken. You guys want to get the next one rolling. Alrighty good luck. Jason Hudley for my Eagles. 33-75 nice color. Eagles having a great break. Lamar Jackson. Ah there is James Robinson. Yes there is. We have Coralyn Sutton. Little tri-color to 199. T. Higgins premier level silver. And we got a field lover base. Joe Reed. Cam Akers-Riggy. Denzo Mims. Troy Aikman. And Christian McCaffrey to 99. Henry Ruggs. Anderson Wens. And look at that Chase Young to 199 coming up. That is a patch autograph for the Washington football team going to Tyler. There you go. Jerry Rice. And there you go Brandon speaking of James Robinson. There he is. Little silver. Mimsie. Phillip Rivers. Can't have a good break without Michael P. Ryan. Very nice color match. This dude kills it for the rest of his career. 17 out of 35. I really mean that. We have the biggest fan club of Michael P. Ryan Adams here at Jaspy's. Or I should say some Jaspy customers on IG. Phillip Rivers. 19 out of 49. Deandre Swift. The Sheriff. Paymani. Kurtz. Drew Reeves field level. Zach Moss. Mike Evans. Coke Matt. Chase Claypool. That is a club level silver. Aviska. Claypool again. James. James Winston. And Malcolm Harry. For the Dolphins. Field level. Alrighty guys and there you go. That was the break. What a great three boxes. We got three green's out of five. Chase Young. Patch Autograph in this last one. Tideye. Patch Autograph. Chase Claypool. Lot of lots of pennies here of course guys. Let me top up some of these bigger hits and then we'll go through a quick little recap. But not too shabby guys. The next. You know what guys I don't even think I see any Herbert in this one. Unfortunately. Not even a base. Which I know is a hefty price for them. But. Maybe it's still hiding in the remaining half case. We had a Chase Young. Patch Autograph. Chase Claypool. Three colored patch. Tideye to 25. Josh Jacobs. We had a one of one. Zach Moss. Dye cut. C.D. Lamb. Field level. Jerry Judy Silver. Jalen Hurts. Patch Autograph. Rugged Silver. Colkmet Autograph. Kylin Murray Green out of five. Jalen Hurts color. Out of 149. C.D. Lamb Green out of five. Silver Jalen Hurts. And Dan Marino to 35. So I appreciate it guys. This is break number 18. Number 19 and 20 or number 19 is in the store. We've only pulled I think one XRC so far too as well guys. We'll see a couple more in the remaining case. So let's get it going guys. JasperysCaseBreaks.com