 In this video, we're going to consider a specific example of a linear transformation. So consider the following map. T will be a map from R3 to R2. So remember, R3 here denotes the vector space of column vectors, which contain three real numbers. And likewise, R2 would be the vector space of column vectors containing exactly two real numbers. So we talked about previously that R3 and R2 are vector spaces. They're some of our favorite vector spaces. We're going to define a function from R3 to R2 by the following rule, T of x1, x2, x3. So these right here are just the generic positions of a vector in R3, right? So R3 just consists of vectors with three real numbers. Let's generically call them x1, x2, x3, right? So T of x1, x2, x3, this is going to equal a vector in R2. The first coordinate will be x1 plus 2x2 and x3, sorry, the second component will be x3 minus 3x2 right there. Now you'll notice, this is actually an important observation I want to make here. You'll notice that the components of T of x, right? This right here, this x1, x2, x3, this is the vector x, right? It's a vector x arrow right there. This is supposed to just be a generic vector in R3. Or if you want to see this way, T of x right here, T of x, this is just supposed to be some, this is just some generic vector in R3 and then this will produce right here a vector in R2. So this is a vector in R2 right there. You'll notice that this, the components of T of x, you look at this right here, x1 plus 2x2, I mean, literally x1 and x2 are numbers, right? But if you kind of pretended they were vectors for a second, x1 plus x2, this looks like a linear combination of the original variables x1 and x2, right? This, the first component of T of x looks like a linear combination of the components of x. And likewise, when you look at the second component of T of x, that also looks like a linear combination of the components of the original vector x. And that's actually not a coincidence. It turns out that it can be shown that a function of the form T goes from fn to fm, this is gonna be a linear transformation if and only if the components of T of x are in fact linear combinations of the components of x, like we see in this example right here. This isn't something we're actually gonna show explicitly, but this is a true fact. We're actually gonna show that, well, we can leave that on the screen. We're gonna show that this is a linear transformation using the definition of the linear transformation. So we saw this on a previous video. A function is a linear transformation if it preserves vector addition and scale and multiplication. So that's our first task here, show T is linear. So the first thing we wanna show is that we're gonna take two generic vectors, we're gonna take x and y as generic vectors in R3. And so what this means here is x would have something of the form x1, x2, x3. And y would likewise have the form, say like y1, y2, y3. So we wanna show that T of x plus y, so that's what we start off with. I'm gonna switch the color to emphasize what we're doing here. We're gonna get T of x plus y. So we have to take this and show that this is equal to T of x plus T of y. So at some point later on, we wanna show that this is equal to T of x plus T of y. So let's unravel what T of x plus y even means. Well, we'll first start off with x plus y. The vector x, like we mentioned above, this is just x1, x2, x3. The vector y just means y1, y2, y3. That's what we said earlier. And so how do you add two vectors together? You just add together their components. Don't forget the T there. So the first component of the sum will just be x1 plus y1. We add together the first components, we add together the second components, and we add together the third components. None of that actually uses the transformation. Now at the next step is when we can actually use the transformation. So remember, what does this transformation do? You are going to take the first component and add to it two times the second component. So we're gonna take the first component and add it to two times the second component of our input vector. So that would look like the first component is x1 plus y1. The second component times, that by two is gonna be x2 plus y2. That's what this transformation does. The first component is the combination of the first one plus two times the second one. Well, what does the second part of the transformation do? The second component is gonna be the third component of the original vector minus three times the second one. All right, so we can do that. We're gonna take this friend right here and subtract from it three times this friend right here. So we end up with x3 plus y3 minus three times x2 plus y2. That is what the transformation does. So now let's come back to where we want it to end up with, because sometimes when you're trying to prove an identity, it's like if you're in a trigometry class, if you have to prove a trig identity, you have the left-hand side, you have the right-hand side. You know where you're starting from and you know where you want to go, but how do you actually get there? Well, part of the journey is you go from the start and you go as far as you know how to go, but you can also work backwards at times. So giving myself just a little bit more space here, I'm just gonna copy this one down here and work backwards. What does t of x mean? t of x means you're gonna take t of x1, x2, x3. Right, what does t of y mean? That means t of y1, y2, y3. That's the definition of x and y here. What is t of x1, two and three? Remember, our transformation will take the, for the first component of the image, you take the first component of the input plus two times the second component. For the second component of the output, you take the third component minus three times the second component. This is t of x. You're gonna add to that t of y, which is y1 plus 2y2, y3, minus three times y3 and cram it in there. Oh yeah, that's super slick. We made it in there. So that's what those things are. And that these are vectors, we can add them together by the usual rules of vector addition. And so if we add these things together, you'll end up with, let me put it here, you're gonna get x1 plus y1 and you're gonna get, so kind of put those friends together. Well, I'll come back to that. And you're gonna get a 2x2 plus 2y2, like so. And then for the second component, you're gonna get an x3 plus y3 and you're gonna get a minus 3x2 minus 3y2, like so. And notice the order I put them in. You have an x1 plus y1. If you factored out the two, you notice here you have a 2x2 plus 2y2. If you factor out the two, oh, that's exactly what you have right there. And then look at the second component. You have an x3 plus y3 and you have this negative 3x2 minus 3y2. If you factored out the negative 3, you would get exactly what we have right here. Oh, so Bob's your uncle. These two things are actually equal to each other. And so now we've established that, we've now established in this example that the transformation preserves vector addition. We took generic vectors. We started off with two vectors. The only thing we knew about x and y is that they belong to R3. And we showed by the formula of the function that it preserves vector addition. That T of x plus y is equal to T of x plus T of y. So it preserves vector addition. What about scalar multiplication? That's the next thing that one would have to show. For scalar multiplication, we're gonna start off with the generic vector. We're gonna take x again, right? Take c to be some scalar times it by x. Well, what does x mean? x is just a vector in R3. So we have x1, x2, x3. Like we saw before. If you scale that by c, that just means times each of the components by c. So you get cx1, cx2, cx3, like so. Now apply the definition of the function T. Remember, T will take the first component, cx1, plus two times the second component. That's the first entry. The second entry will be the third component minus three times the second component. Like so. Notice that in the first entry, there's this common factor of c. In the second entry, there's a common factor of c. You could factor these things out. You get c times x1 plus 2x2. You get c times x3 minus 3x2. And again, notice how c is divisible in every entry of the vector. So we can factor it out via scalar multiplication. This will then give us c times x1 plus 2x2, x3 minus 3x2. And what is this right here? This vector is just T of x. So this gives you c times T of x, like so. And so we now see that T of cx is equal to c T of x. And that's it. We've now established that this transformation is a linear transformation. This map coming back to the top, it preserves both vector addition and scalar multiplication. This rule right here. And so this shows that is in fact, a linear transformation. And this is why I said that if your transformation just writes the output as, it's like if the output variables are just linear combinations of the input variables, it will always be a linear transformation because the template you now see in front of you would work for every single such function like that. And you'll have some opportunity to play around that in the homework. Of course, if you are doing the homework, I certainly hope you are. And so we can see that this is a very nice way of constructing linear transformations. And we'll see of course in the future that essentially this is the only way one can construct a linear transformation. But it is important very much as we study linear transformations that we're able to recognize what a linear transformation is and what it's not. A linear transformation is exactly those transformations that preserve vector addition and scalar multiplication. And it's important that we're able to prove such a thing.