Transforming 3D and 4D vectors with matrices
Using matrices to transform a 4D vector
- [Trainer] We've already thought a lot about two by two transformation matrices as being able to map any point in the coordinate plane to any other point or any two-dimensional vector to any other two-dimensional vector. What we're going to do in this video is generalize a bit and realize that the same principles can be used for n dimensional spaces. Now I know that sounds a little bit fancy and it really is on some level, but it's really the same ideas. So for example, let's extend what we know about two dimensions, let's extend it to say four dimensions. So let's write a four-dimensional vector here and it is hard to visualize in four dimensions, so don't be hard on yourself if you have trouble. Two dimensions not too hard, three dimensions not too hard, four dimensions a little bit hard for us. Maybe we have to think about time as the fourth dimension but in matrix world or in vector world, it's pretty easy to represent them as hard a it is to visualize. So a four-dimensional vector, we'll just have four numbers. Negative one, let's see, negative three, I'm just making these up randomly, negative five and one. This is a four-dimensional vector, and we could view it as being a weighted sum of the unit vectors in the different dimensions of four-dimensional space. I guess you could say it. You could say that this is the same thing as, actually let me color code a little bit. This would be equal to negative one times the one, zero, zero, zero vector plus negative three, plus negative three times the zero, one, zero, zero vector plus negative five, plus negative five times the zero, zero, one, zero vector. I think you see where this is going. And then last but not least, plus one times the zero, zero, zero, one vector. Now when I write it this way, you might immediately start realizing, "Oh I think I know how to do transformations here." For example, if I were to give you the transformation matrix and this would be a transformation matrix for four dimensions. This is gonna be a four-by-four matrix. So I'm gonna write some random numbers here. One, zero, negative three, negative one, two, zero, negative three, one, three, two, zero, two, three, negative one, zero and three. So my question to you is, what would be the mapping of this four-dimensional vector if we were to apply this transformation to four-dimensional space? What would be the result? Pause this video and think about it. Well, it's completely analogous to what we did in the two-by-two world in two-dimensional space. We thought about, all right, instead of the one, zero, zero, zero vector, we're now going to use this vector. Instead of the zero, one, zero, zero vector, we're now going to use this vector. Instead of this one in that blue-green color, we're now going to use this one. And last but not least instead of that, I guess we could say same in colored vector, we're now going to be using this one. So another way to think about it is, the mapping of this vector, let me write it this way. Let me make a little line here so we can separate things a little bit but we could write, all right, a little bit smaller, hopefully you can see this. So this is our original vector, negative five, one, but we wanna do the prime. What does it get mapped to under this transformation? Well, this is going to be negative one, instead of this unit vector right over here, it's gonna be negative one of this one right over here. So it's negative one times all of this business, one, two, three, and three. And then we could have just instead of plus negative three I can just write in minus three times all of this business, zero, zero, two, negative one. And then we have minus five times all of this business, negative three, negative three and then we get zero, zero, and then, that definitely gets a little bit more work involved, the more dimensions we have, plus one times this business. So plus one times negative one, one, two, three. And so what's this going to be equal to? So actually this could be a good time to pause the video too and have a go at it. All right, so this is going to be this first one, I just make all of these negatives. So negative one, negative two, negative three, negative three, and to that, I'm going to add, let's see if I multiply all of those times negative three, I'm going to get zero, zero, negative six and positive three. And then if I multiply all of these times negative five, I am going to get 15, 15, zero and zero. And then if I multiply all of these times one, well, I just get those things again. So that's going to be negative one, one, two, and three and we are in the home stretch. So now we can just add everything together the corresponding terms. And so this is going to be negative one plus zero, plus 15, plus negative one. So that's gonna be the same thing as 15 minus two which is going to be 13, then negative two plus zero, plus 15, plus one, so that's gonna be 16 minus two which is 14, then we have negative three plus negative six which is negative nine and then we add two to that, so that is negative seven. And then negative three plus three is zero, plus zero is zero, plus three is three, and we are done. We have found the mapping of this four-dimensional vector based on a four-by-four transformation matrix. Very cool.