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MOOC_LinearAlgebra_Lesson04

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Hello, and welcome to lesson four of this Introduction to Linear Algebra with Wolfram U. The topic for this lesson is Vectors in N Dimensions. Let's begin with a brief overview of the lesson. what’s a vector? You can think of a vector as being an object, which has both magnitude, like a length and a direction. Here are two vectors, one of them is blue, the other one is red. And they each have a length, which happens to be different in this case. And they have a direction given by these arrows. In linear algebra you think of vectors differently. You think of them as being ordered list of numbers. Each vector over here is really going to be an ordered list of numbers. On the other hand, a scalar is simply a number, it could be positive, or negative, or zero. Now, you can combine these vectors by using a combination of multiplication by scalar in addition, and we get the idea of a linear combination. given another vector does it have the property that it can be expressed as a linear combination of the given vectors? That leads to the notion of vector equations and linear systems. So that's the plan for this lesson talk about vectors and vector equations. Let's begin with a simple case of vectors in two dimensions. Vector in two dimensions is just a list of two numbers. For example, you got a vector, 3 and -1. It's got two entries over here. Let’s define it, like MatrixForm to it, you get back a column vector. You think of these vectors as being really columns of a matrix. They’re column vectors and if you put them all together, then the set of all vectors in two dimensions has got a special notation, it’s called R^2. That’s written as double struck capital R, it’s a special symbol, a mathematical symbol, which is worth knowing. when are such vectors equal? They’re equal if the corresponding entries are equal. For example, the two vectors over here are not equal, because the first 1 is 4, 7, the other one is 7, 4. Let’s check that over here. If you ask the question, is v=w, using the double equal sign, you get back a false, which says that those vectors are not equal. Now let’s try and combine the vectors in two dimensions. There are two operations the scalar multiplication, namely multiply every entry by a scalar, and there’s addition, simply add the corresponding increase. For example, you’ve got two vectors u and v over here. u is 1-3, v is 2-7. The question is: find 3u, find -5v, find u+v and find 2u-3v. Let’s first define them and then what you can do is, you can work out 3*u, just multiply each entry by 3, so you get 3, -9. You can do -5v, just multiply each entry by -5 for v. You can add the entries to get u+v. The last one is a bit more difficult because you really should do 2u, then do 3v, then subtract them, but in fact, it's quite easy to do in the Wolfram Language. So that's how we combine vectors using scalar, multiplication, and addition. Now of course, because we're in two dimensions, one can talk about the geometry of vectors in two dimensions. What you can do first of all, is set up a rectangular coordinate system in the plane, and then a vector is simply going to be the same as a point. For example, over here you have a set of points on the left and a set of vectors on the right, and each point over here corresponds to a vector over there. If you’re given two vectors, u and v, then u+v really corresponds to the fourth vertex of a parallelogram. In this picture over here, you’ve got a parallelogram with four vertices. There's the origin, there's u, there's v and then there’s u+v. Those are the vertices of the parallelogram. Having defined vectors in two dimensions, let’s get a little ambitious and talk about vectors in three dimensions. A vector in three dimensions is just a column vector, column matrix with three entries. Like before, they are defined geometrically by points, with arrows from the origin. For example, here's a vector in three dimensions, 2, 3, 5, that’s a. Then 2a is just a vector along the same direction but twice as long. Having talked about two and three dimensions, we can now go ahead and talk about vectors in n dimensions, where n is any positive integer. Could be 2, could be 3, could be 10, whatever, and that's the space R^n, a very important mathematical construct. So R^n denotes a set of all vectors in n dimensions. You can think of each vector over here as being a collection of scalars u_1, u_2, to u_n. It's just a column vector with n entries. The special vector called a zero vector, which is just denoted by zero. Again, you can combine vectors using scalar multiplication or using addition. The set of all these vectors with these operations is referred to as the vector space R^n. Vector spaces are just collections of all vectors for a certain type, with the operations defined by scalar multiplication and addition, and the most important vector space for this course is the space R^n. A simple example, here is a pair of vectors in four dimensions, u and v. You can combine them, you can do u+v and you can do 3u-7v, and it's all quite simple. That's the idea of a vector space and now we can talk about linear combinations. Suppose you’re given a fixed set of vectors, v_1, v_2, up to v_n, n could be 3 or 4, or whatever, and scalar c_1, c_2, up to c_n. Then what you can do is, you can combine them. You can first multiply each vector by the corresponding scalar and then you can add them up and you get a linear combination of those vectors, with the weights c_1, c_2, up to c_n. For example, here are the same vectors u and v from the earlier slide, and you can let’s say do √3*u+v, which just says multiply each entry of u by square root of three, then add it to v, and you get back the result over here. Or you could do ½*u and 0*v, that just gives you back one half of u. Or could do zero times each vector, which of course, gives you just zero as a result. That's about fixed vectors and fixed scalars but what would happen if you actually took a fixed set of vectors over here and then try to find all the linear combinations. Well that's called the span of those vectors. Basically you take the collection of all vectors, which are linear combinations of this particular set of vectors. Now that looks really abstract but for example, if you have a single vector v, then that’ll give you a line through the origin. For example, use a vector v and you multiple it by c’s and you get back a straight line. Then on the other hand, if we’ve got a pair of vectors, u and v, then you get back a plane because you couldn't do all the multiples of u over here, all the multiples of v over here, and then you can add them up and get back a plane. So that's the notion of a span or subset. suppose you’re given an arbitrary vector, which is not one of those, suppose you're going to vector b, can you somehow combine them as a linear combination of the views? Here's your v_1, v_2, given to you and it's a b. can you actually check whether b belongs to the span of those vectors or not? A span in this case is just a plane. You want to figure out whether b is in that plan or not. The question is: does this, over here, have a solution or not. To do that, what you do is set up the augmented matrix for the system. You've got 1phi, -3, 2, -13, 8, and 3, 3, 2, because you got 1, 5, -3 over here, 2, -13, 8 over there, and 3, 3, 2 over there. What you do is you do a row reduce. Then you see over here that the last row is 0=1, that’s no good. That says that this system over here has got no solution, and hence, the given vector b is not in the span of v_1 or v_2. Somehow starting off with these spans will come to linear systems and vector equations and that's how we determine if a particular vector b belongs to span or not. That brings me to the end of lesson. The point is in this lesson, you've learned how to represent vectors in n dimensions, not just 2 or 3, but any dimensions, and they are just column vectors. We also talked about vector spaces. The collection of all vectors in R^n and linear combination of vectors. And then given a subset of vectors, you can talk about the span. Then the question is: how does that relate to linear combination? That's just a collection of all linear combinations and that leads to the question of a vector equation, every time you want to check whether a vector b belongs to the span or not. That's the end of this lesson and in the next lesson we’ll talk on matrix equations, but before we do that, do this important lesson, it's quite abstract. Do go over it carefully and be ready for matrix equations. I’ll stop over here. Thank you very much.

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Posted by: wolfram on Sep 30, 2020

MOOC_LinearAlgebra_Lesson04

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