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Hello, and welcome to Lesson 4 of this Introduction to Linear Algebra with Wolfram U. The topic for this lesson is vectors in <i>n</i> dimensions. Let's begin with a brief overview of the lesson.  The question is: 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. But in linear algebra, you think of vectors slightly 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 scalars and addition, and we get the idea of a linear combination. The final question is: 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. OK, so, let's begin with a simple case of vectors in 2 dimensions. A vector in 2 dimensions is just a list of two numbers. For example, here you've got a vector, 3 and -1. It's got two entries over here. Let's define it. You apply 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 2 dimensions has got a special notation; it's called <b>R</b>^2. That's written as a double-struck capital R; it's a special symbol, a mathematical symbol, which is worth knowing. The question is: when are such vectors equal? Well, they're equal if the corresponding entries are equal. So for example, the two vectors over here are not equal, because the first one is {4, 7}, the other one is {7, 4}. Let’s check that over here. If you ask the question, is <i>v</i> equal to <i>w</i>, using the double equals sign, you get back a False, which says that those vectors are not equal. Now, let's try and combine the vectors in 2 dimensions. There are two operations: the scalar multiplication, namely, multiply every entry by a scalar, and there's addition, simply add the corresponding entries. For example, you've got two vectors <i>u</i> and <i>v</i> over here; <i>u</i> is {1, -3}, <i>v</i> is {2, -7}. The question is: find 3<i>u</i>, find -5<i>v</i>, find <i>u</i> + <i>v</i> and find 2<i>u</i> - 3<i>v</i>. Let's first define them, and then what you can do is you can work out 3<i>u</i>; just multiply each entry by 3 so you get {3, -9}. You can do -5<i>v</i>, just multiply each entry by -5 for <i>v</i>. You can add the entries to get <i>u</i> + <i>v</i>. The last one is a bit more difficult because you really should do 2<i>u</i>, then do 3<i>v</i>, then subtract them, but in fact, it's quite easy to do using the Wolfram Language. So that's how we combine vectors using scalar multiplication and addition. Now, of course, because we're in 2 dimensions, one can talk about the geometry of vectors in 2 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, <i>u</i> and <i>v</i>, then <i>u</i> + <i>v</i> 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 <i>u</i>, there's <i>v</i> and then there's <i>u</i> + <i>v</i>. Those are the vertices of the parallelogram. Having defined vectors in 2 dimensions, let's get a little ambitious and talk about vectors in 3 dimensions. A vector in 3 dimensions is just a column vector, a 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 3 dimensions, {2, 3, 5}, that's <i>a</i>. Then 2<i>a</i> is just a vector along the same direction but twice as long. Having talked about 2 and 3 dimensions, we can now go ahead and talk about vectors in <i>n</i> dimensions, where <i>n</i> is any positive integer, could be 2, could be 3, could be 10, whatever, and that's the space <b>R</b>^<i>n</i>, a very important mathematical construct. So <b>R</b>^<i>n</i> denotes a set of all vectors in <i>n</i> dimensions. You can think of each vector over here as being a collection of scalars <i>u</i>_1, <i>u</i>_2, up to <i>u</i>_ <i>n</i>. It's just a column vector with <i>n</i> entries. There's a special vector called a zero vector, which is just denoted by 0. 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 <b>R</b>^<i>n</i>. Vector spaces are just collections of all vectors of a certain type, with the operations defined by scalar multiplication and addition, and the most important vector space for this course is the space <b>R</b>^<i>n</i>. A simple example: here is a pair of vectors in 4 dimensions, <i>u</i> and <i>v</i>. You can combine them— you can do <i>u</i> + <i>v</i> and you can do 3<i>u</i> - 7<i>v</i> 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, <i>v</i>_1, <i>v</i>_2, up to <i>v</i>_<i>m</i>, <i>m</i> could be 3 or 4 or whatever, and scalars <i>c</i>_1, <i>c</i>_2, up to <i>c</i>_ <i>m</i>. 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 <i>c</i>_1, <i>c</i>_2, up to <i>c</i>_ <i>m</i>. For example, here are the same vectors <i>u</i> and <i>v</i> from the earlier slide, and you can, let's say, do √3 • <i>u</i> + <i>v</i>, which just says, multiply each entry of <i>u</i> by √3 then add it to <i>v</i> and you get back the result over here. Or you could do ½ • <i>u</i> and 0 • <i>v</i>; that just gives you back ½ <i>u</i>. Or you could do 0 times each vector, which, of course, gives you just 0 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 tried 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 <i>v</i>, then that'll give you a line through the origin. For example, here’s a vector <i>v</i>, and you multiply it by <i>c</i>'s and you get back a straight line. And on the other hand, if we've got a pair of vectors, <i>u</i> and <i>v</i>, then you get back a plane because you can do all the multiples of <i>u</i> over here, all the multiples of <i>v</i> over there, and then you can add them up and you get back a plane. So that's the notion of a span or subset, and the question now is: suppose you're given an arbitrary vector, which is not one of those. And the question is: suppose you're going to vector <i>b</i>, can you somehow combine them as a linear combination of the <i>v</i>'s? For example, here's your <i>v</i>_1, <i>v</i>_2 given to you, and that's a <i>b</i>. The question is: can you actually check whether <i>b</i> belongs to the span of those vectors or not? A span in this case is just a plane. You want to figure out whether <i>b</i> is in that plane or not. The question is: does this have a solution or not? To do that, what you do is you set up the augmented matrix for the system. You've got {1, 5, -3}, {2, -13, 8}, and {3, 3, 2}, because you've 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 <i>b</i> is not in the span of <i>v</i>_1 or <i>v</i>_2. Somehow, starting off with these spans, we have come to linear systems and vector equations, and that's how we determine if a particular vector <i>b</i> belongs to a span or not. That brings me to the end of the lesson. The point is, in this lesson, you've learned how to represent vectors in <i>n</i> 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 <b>R</b>^<i>n</i> and linear combinations 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 combinations? Well, that's just a collection of all linear combinations and then that leads to the question of a vector equation, every time you want to check whether a vector <i>b</i> belongs to the span or not. That's the end of this lesson, and in the next lesson we'll talk about matrix equations, but before we do that, do 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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Duration: 10 minutes and 53 seconds
Language: English
License: Dotsub - Standard License
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Views: 9
Posted by: wolfram on Sep 30, 2020


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