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MOOC_LinearAlgebra_Lesson01

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Hello and welcome to Introduction to Linear Algebra with Wolfram U. My name is Devendra Kapadia and I'm the Manager of Calculus and Algebra at Wolfram. But I also enjoy teaching mathematics, and, in fact, I have taught mathematics at high school and college level for many years. It is a great pleasure to share my ideas about linear algebra with you in this course. This is the first lesson of the course, let’s begin by asking the question: what is linear algebra? Well, linear algebra means different things to different people. For some people linear algebra means linear systems. Here is a linear system of equations. You have the equations x+y=5 and x-y=1. And there, of course, is just a pair of straight lines in the plane, which you can see graphed over here. Given the system, we have some key questions to ask. For example, does this system have a solution or not? Now, in this simple case, of course, we have that x is three, and y is two, and that's the solution. But in some other cases, you may not really have a solution at all, or it might be quite hard to find a solution. So, we have a notion of an approximate solution. So, either you have an exact solution or you have an approximate solution. At any case, for some people, their definition of linear algebra is, linear algebra is the study of linear systems of equations. But other people have a more geometrical view of a subject and they’ll see that linear algebra is the study of geometric transformations. For example, here in this figure you have a reflection about the line y=x in the plane. And this, of course, is a linear transformation because the image of a line is, again, a line and given such a transformation, you could ask: find an algebraic description of this transformation, or you could ask: are there any lines that are left unchanged by the transformation? For example, over here you can see that the line y=x is kind of invariant or unchanged for this transformation. So, some people who tell you that linear algebra is a study of linear geometrical transformations. There is a third viewpoint, which is quite popular these days, and that's the notion of matrices and determinants. So matrix is just a rectangular array of numbers in rows and columns. So here's a matrix, it's got two rows. First row is 5, 4, second row is 6, 7. And 2 columns, so that's a 2 by 2 matrix. A system of linear equations and transformations can actually be represented using matrices. Now, a matrix is a collection of numbers, that's a huge collection typically but, with every matrix you have its determinant. The determinant is just a number that's associated with the matrix. For example, in this case, you have the determinant as 11. That comes out by doing 5*7=35, 6*4=24, subtract them and get 11, and this determinant, although it’s just a number, it encodes, it contains important information about the system of equations or the transformation that you’re studying. So in the modern world, a nice way to think of linear algebra, is that it’s a study of matrices and their determinants. And of course, that's useful, because in this day of computers we can store matrices quite easily with powerful computers. So, doing linear algebra is actually a very fruitful way of solving problems in the modern world. Now of course, there's a long history behind the subject but let me just give you a brief history of the subject of linear algebra, as it relates to this course. So, of course, the Chinese and other civilizations knew about linear algebra. But to me, the course has history going back to roughly 1750 when Gabriel Cramer published a study of determinants. And in fact, he published a rule for solving systems of equations. And a little later, 1800, the famous German mathematician Carl Gauss actually talked about elimination to solve equations and the least squares method, which we'll talk about later in the course. Now, Gauss was a genius but his real interest in this case, was to try and do geodesy and apply these ideas to real world problems. But the mathematical theory of linear algebra really began around 1850, when James Sylvester use the term matrix or “womb” for an array of numbers. Sylvester and his friend, Arthur Cayley, worked on matrices, and Arthur Cayley actually published his work on the theory of matrices around 1855. Sylvester and Cayley were great friends. In fact in those days, it was hard to make a living doing mathematics. So Sylvester was an actuary, he worked in insurance. Cayley was a lawyer, and they talked about law in the day, and talked about mathematics in the evening. Together, they set up a lot of what we call Linear Algebra Theory. A little later, in 1888, Giuseppe Peano actually talked about vector spaces. In the modern mathematical world, vector spaces are very important, but in this course we will not worry too much about them, we’ll mostly work with matrices and determinants as taught to us by Cayley and Sylvester. what happens today? Well today, people all over the world use linear algebra in different ways. In different professions there are their own uses for linear algebra. For example, if you do mathematics, you might want to study the geometry of surface, and then you’d use linear algebra to actually make a system of coordinates in which you can do your matrix manipulations. Engineers use linear equations to study the motion of fluids, to design airplanes and such things. Data scientists will have their own linear models to make predictions about data. In fact, our data science is a big driver for linear algebra these days. Now, if you study physics, then physicist will use linear algebra to study quantum mechanics, the study of atoms. In fact, these days, they also use linear algebra to build quantum computers. And finally, it's not just the sciences. Economists have always used linear algebra for doing things like, input-output matrices, and in fact, a lot of modern linear algebra came around because of these applications. So not only have the applications used linear algebra, but they actually have driven the modern development of the subject. So linear algebra is very much an applicable subject in the modern world. Now, of course, you might ask the question: why should I study linear algebra? Well, I'll give you a few reasons for studying linear algebra. The first of all, it has very broad applications in the real-world. We just saw that. If you apply for certain kinds of stem degrees, it's an essential course which you'll need to cover at some point. In fact, in some cases, they may even ask you, how good is your linear algebra for advanced degrees. Now, even if all that were not true, I would say that, learning linear algebra is a great thing, because linear algebra is a major intellectual endeavor, and to study is a major intellectual achievement. So, I’d say that to study linear algebra is to master one of the most important branches of modern mathematics. And I strongly urge you to do it with this course. So now let me tell you a little bit about the course itself. The main goal of this course is to give you a thorough introduction to linear algebra. And to do that, I'll begin by talking about linear equations and linear transformations. And then we'll talk about matrices and the determinants and then we'll move on to more advanced topics like eigenvalues and other things. Now, to make sure you understand what's going on, we will have excellent quizzes going on to accompany the lessons, and as far as proofs are concerned, there will be no formal proofs in the course and we'll focus on understanding the subject without any proofs. Now, my main goal in all this is to show you the linear algebra's really easy, and I hope that when you're done, you will feel that you've learned linear algebra in a very useful way. And you’ll be able to apply it in your own work. Whatever subject it is that interests you. who is this course for? Well, you might be an ambitious middle schooler or high schooler who wants to learn advanced mathematics, become an engineer, become a data scientist, whatever. I would say to you that, to study linear algebra is a great way to start learning more mathematics. So this course is really designed for people who are learning linear algebra for the first time. Or you might be a busy professional, who needs to get some knowledge of linear algebra to, let’s say, take a course on machine learning or whatever, and then this course should be a nice refresher for you. I think that this course is good for anyone who wants to know how the Wolfram Language can be used to learn and do linear algebra. So, if you love mathematics and you love programming, then this course is certainly for you. Whether you are a student, a professional, a teach, or anyone else. I think this course is meant for a wide variety of people and I’ve tried to keep it as simple as possible. That brings me to the end of this introduction. To summarize, linear algebra is a study of linear equations and geometric transformations. The modern approach to the subject will use matrices, determinants. and of course, at a more advanced level, it will also use the vector spaces. Now, to know linear algebra is important because it's an essential tool for students in many fields. That's the broad introduction to the course. In the next lesson will begin the course formerly with an introduction to linear systems. Thank you very much. I hope you enjoyed this introduction. I do hope you'll pursue this course with great sincerity and wish you good luck. Thank you.

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Duration: 11 minutes and 36 seconds
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Language: English
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Posted by: wolfram on Sep 30, 2020

MOOC_LinearAlgebra_Lesson01

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