WEBVTT mathematics/linear-algebra/hovasapian
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Welcome back to educator.com and welcome back to linear algebra.
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In our last lesson, we talked about the diagonalization of symmetric matrices, and that sort of closed out and rounded out the general discussion of Eigen values, Eigen vectors, Eigen spaces, things like that.
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The Eigen value and Eigen vector problem is a profoundly important problem in all areas of mathematics and science, especially in the area of differential equations and partial differential equations in particular. It shows up in all kinds of interesting guises.
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Of course, differential equations and partial differential equations are pretty much the, well, it is what science is all about, essentially, because all phenomena are described via differential and partial differential equations.
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So this Eigen vector, Eigen value problem will show up profoundly often.
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Today we are going to talk about linear mappings, and the matrices associated with linear mappings. Mostly just the linear mappings. We will get to the matrices in the next couple of lessons.
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But, so you remember some lessons back, we actually talked about linear mappings, but we mostly talked about them from... like a 3-dimensional space to a 4-dimensional space.
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RN to RM, some kind of space that we are familiar with for the most part. But, you also remember we have used examples where a space of continuous functions is a vector space, the space of polynomials of a given degree is a vector space...
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So, these vector spaces, they do not, the points in the vector spaces do not have to necessarily be points the way that we are used to thinking about them, they can be any kind of mathematical object if they satisfy the properties of a vector space.
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Well, vector spaces are nice, and we like to have that structure to deal with, but really what is interesting... when linear algebra becomes interesting is when you discuss mappings, and in particular linear mappings between vector spaces.
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Now, we are going to speak more abstractly about vector spaces in linear mappings, as opposed to specifically from RN to RM.
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Much of this is going to be review, which is very, very important for what it is we are going to be doing next. So, let us get started.
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Okay. Let us start off like we always do with a definition. Let us go to black. I have not used black ink in a while.
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Definition, let v and w be vector spaces, a linear mapping, which is also called a linear transformation, L(v) into w, and this into is very important for our definition.
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We will talk about it more why we choose into here and another word later... into is a function, a signing... unique vector that we signify as L(u) in w to each vector u in v, such that the following hold... a, and if I have 2 vectors -- u and u... u and v -- if I have two vector u and v, both of them in v, and if I add them, then apply the linear mapping, that is the same as applying the linear mapping to each of them separately and then adding them... for u and v in v.
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The second thing is... if I apply, if I take some vector u and I multiply by a constant then apply the function, it is the same as if I were to take the vector alone, apply the linear function and then multiply it by the constant.
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For u and v... and k is any real number. Okay. Let us stop and take a look at this really, really carefully here.
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This -- let me use a red -- this plus sign here on the left, this is addition in the space v. Let me draw these out so you see them.
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That is v, this is w, my linear map is going to take something from here, do something to it, and it is going to land in another space, okay?
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So, this addition is -- here let me... v... w... -- this addition on the left, these, this is addition in this space... a vector u and v, they are here... I add them and then I apply.
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This addition over here, this is addition in this space. They do not have to be the same. It is very, very important to realize that. This is strictly symbolic.
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As you go on in mathematics everything is going to become more symbolic and not necessarily hvae the meanings that you are used to seeing them with.
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Yes, it is addition, but it does not necessarily mean the same addition.
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So, for example, I can have R3, where I add vectors, the addition of vectors is totally different. A vector plus a vector is... yes, we add individual components, but we are really adding a vector and a vector, two objects.
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This might be the real number system where I am actually adding a number and a number. Those are not the same things, because numbers are not vectors.
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So, we symbolize it this way as long as you understand over on the left it is the addition in the departure space, over here on the right it is addition in the arrival space.
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Okay, so, let us talk about what it is this means... if I have a vector u and if I have a vector v, I can add them in my v space, I stay in my v space and I get this vector u + v. It is another vector.
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It is a closure property, it is a vector space. Then, if I apply the linear mapping L to it, I end up with some L(u+v). That is what this symbolizes. I do something to it, and I end up somewhere.
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In order for this to be linear, it says that I can add them and then apply L, or I can apply L to u to get L(u), and I can apply L to v separately to get L(v), and now when I add these, I end up with the same thing.
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Again, this is extraordinary. This is what makes a linear mapping. Okay? It has nothing to do with the idea of a line. That is just a name that we use to call it that. We could have called it anything else.
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But, it actually preserves a structure moving from one space to another space. There is no reason why that should happen, and yet there it is. We give it a special name. Really, extraordinarily beautiful.
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Okay. If the linear mapping has to be from a space onto itself, or a space onto a copy of itself... in other words R3 to R3, R5 to R5, the space of polynomials to the space of polynomials, we have a special name for it... we call it an operator.
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Let me separate these words here... We call it a linear operator.
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Operator theory is an entirely different branch of mathematics unto itself. Operators are the most important of the linear mappings, or the most ubiquitous.
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Okay. Let us recall a couple of examples from linear operators -- of linear maps, I am sorry -- recall our previous examples of linear maps.
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We had something called a projection... the projection was a map from, let us say, R3 to, let us say R2.
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Defined by L of the vector (x,y,z), take a three vector and I end up spitting out a 2 vector.
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I just take the first 2 (x,y), it is called a projection. We call it a projection because we are using our intuition to name this.
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It is as if we shine a light on a 3-dimensional object, the shadow is a 2-dimensional object. All shadows are two dimensional. That is what a projection is. I am projecting the object onto a certain plane. I am projecting a 3-dimensional object onto its 2-dimensional shadow, creating the shadow. That is a linear map.
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Dilation. This is a linear map. This is actually a linear operator. R3 to R3, and it is defined by L of some vector u is equal to R × u. I am basically just multiplying it by some real number, where R is bigger than 1.
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Dilation means to make bigger. So, I expand the vector.
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A contraction. Contraction is the same thing, so I will just put ditto marks here... and it is defined by the same thing, except now R is going to be > 0 and < 1. So, I take something, a vector, and I make it smaller. I contract it, I shrink it.
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We have something called reflection. L is from R2... this is also a linear operator... I am mapping something in the plane to something in the plane. It is defined by L of the vector (x,y), or the point (x,y) = x - y. That is it. I am just reflexing it along the x axis.
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There is also a reflection along the y axis if I want, where it is the x that becomes negative. Same thing.
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The final one, rotation, which is probably the most important and the most complex... and most beautiful as it turns out... of the linear maps... also a linear operator. R2 to R2, or R3 to R3. We can rotate in 3-dimensions.
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We can actually rotate in any number of dimensions. Again, mathematics is not constrained by the realities of physical space. That is what makes mathematics beautiful.
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These things that exist and are real in real space, they exist and are real in any number of dimensions... defined by L(u), if I take a vector, and if I multiply it by the following matrix, cos(θ) - sin(θ), sin(θ), cos(θ), × the vector u.
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If I take a vector u, and I multiply it on the left by this matrix, cos(θ), -sin(θ), sin(θ), cos(θ), this two by two matrix... I actually rotate this vector by the angle θ. That is what I am doing.
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Every time you turn on your computer screen, every time you play a video game, these linear maps that are actually making your video game, making your computer screen possible. That is what is happening on the screen.
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We are just taking images, and we are projecting them, we are dilating them, we are contracting them, we are reflecting them, we are rotating them... at very high speeds of course... but this is all that is happening. It is all just linear algebra taking place in your computer, on your screen.
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Okay. So, in order to verify that a function is a linear mapping, we have to check the function against the definition. It means we have to check the part a and part b. Okay, so let us do an example.
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Let us see, let v be an n-dimensional vector space... does not say anything about the nature of this space, just says n-dimensional... We do not know what the objects are... space... vector space... n-dimensional vector space.
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Let s = v1, v2, all the way to vN be a basis for v.
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Okay. We know that for v, a vector v in the vector space v, we can because this is a basis, we can write it as a series of constants... c1 × v1, just a linear combination of the elements of the basis. That is the whole idea of a basis. Linearly independent and spans the entire base.
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Every vector in that space can be written as a linear combination. A unique linear combination, in fact... 1 + c2v2 + cNvN.
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Okay. So, let us just throw that out there. Now, let us define our linear map, which takes v and maps it to the space RN, to N-space.
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So some N-dimensional vector space v, and it is going to map to our N-dimensional Euclidian space, RN, and defined by L(v), whatever the vector is, I end up taking its -- I end up with the coordinate vector.
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So, I have some vector v in some random vector space that has this basis. Well, anything in v can be written this way.
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Therefore this v, what it does is it takes this v and it maps it to the RN space, which is the list of coordinates which is just the constants that make up the representation from the basis.
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So, I am taking the vector v, and I am spitting out the coordinate vector of v, with respect to this basis s.
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Okay. Is this linear map -- I am sorry -- is this map linear? Is L linear? We do not know if it is linear, we just know that it is a mapping from one vector space to another.
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Well, let us check a. A, we need to check whether the sum of two vectors in v? Does it equal L(u) individually + L(v).
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Well, let us do it. L(u + v). Well, we just use our definition, we just plug the definition in.
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That is equal to u + v, the coordinate of that. Well, we already know that the coordinates are themselves are linear. So, this is equal to the coordinate of u with respect to s + the coordinate of v with respect to s, but that is just L(u) + L(v).
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So, I have shown that this equals that. So, part a is taken care of. So, now let us do part b.
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We need to show that L(k × u)... does it equal k × L(u)?
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Well, L(k × vector u) = k × u, that is the coordinate vector of ku, but the coordinate vector of k × u with respect to s is equal to k × the coordinate vector of u with respect to s.
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That is equal to k × L(u), because that is the definition. So, we have shown that L of ku equals k × L(u), so yes, b is also taken care of.
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So, this map, that takes any vector from a vector space with a given basis and spits out... does something to it and spits out the coordinate vectors... the coordinate vector with respect to the basis s, which is just the coefficients that make it up, this is a linear map. That is all we are doing, we are just checking the definition.
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Let us throw out a nice little theorem here. Let L from v to w, be a linear map... a linear transformation. Then, a L of the 0 vector in v is equal to the 0 vector in w.
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So, we put this v and w to remind us that we are talking about different spaces. If I take the 0 vector in b, and if I apply L to it, it maps to the 0 vector in my arrival space. That is kind of extraordinary actually.
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And b, which will make sense. It is just the inverse of addition. It says L(u - v) = L(u) - L(v), so we are just extending this subtraction.
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Okay. Now, let us have another theorem, which will come in handy. Let L be a mapping from v to w, and we will let it be a linear mapping of an n-dimensional vector space into w.
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Also, let s = set v1, v2, just like before, all the way to vN, be a basis for v. So, I have a vector space v, I have a basis for v, and I have some function L which takes a vector in v and spits out something in w.
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If u is in v, then L(u) is completely determined... I am going to be a little bit clearer here. Let me actually write out all of my letters. Completely determined by the set L(v1) L(v2), so on and so forth... L(vN).
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I will tell you what this means. If I have a vector space v, and if I have a basis for that vector space... and if I take some random vector in v, and apply the linear transformation to it, I end up somewhere in w. Well, because I have a basis for v, I know exactly where I am going to end up in w. 2138 Because all I have to do is take these basis vectors, v1 to vN, apply L to them, and the L(v1), L(v2), L(v3) all the way to L(vN)... they actually end up becoming precisely the vectors, in some sense, that are needed to describe w, where I ended up. That is what linearity means.
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For this particular theorem, I am not going to go ahead and give you a complete proof, but I am going to make it plausible for you here. So, let us take this vector v, in the vector space v... well, we know that we can write v as c1v1 + c2v2 + so on and so forth + cNvN.
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Okay. Now, let us apply L to this. Well, L(v) is equal to L of this whole thing. c1v1 + c2v2 + so on and so forth + cNvN.
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Well, L is a linear map. That is the hypothesis of the... that is the given part of the theorem. It is a linear map. Well a linear map, just pull out the linearity by definition. That equals c1 × L(v1) + c2 × L(v2) + so on and so forth + cN × L(vN).
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So, again, if I have a basis for my departure space, and I take some random vector v in that departure space, I transform it, you know do some function to it, I already know what my answer is going to be... it is going to be precisely the coefficients c1, c2, all the way to cN multiplied by the transformation on the basis vectors.
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All I have to do is operate on the basis vectors and I stick the coefficients that I got from my original v and I have got my answer. Where I ended up in my arrival space.
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So, again, it is completely determined where I end up in my arrival space is completely determined by the linear transformation on the basis of the departure space.
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Okay. Thank you for joining us at Educator.com for this particular review of linear mappings, we will see you next time.