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Video transcript
I'm going to do one more video where we compare old and new definitions of a projection. Our old definition of a projection onto some line, l, of the vector, x, is the vector in l, or that's a member of l, such that x minus that vector, minus the projection onto l of x, is orthogonal to l. So the visualization is, if you have your line l like this, that is your line l right there. And then you have some other vector x that we're take the projection of it on to l. So that's x. The projection of x onto l, this thing right here, is going to be some vector in l. Such that when I take the difference between x and that vector, it's going to be orthogonal to l. So it's going to be some vector in l. This was our old definition when we took the projection onto a line. Some vector in l. Maybe it's there. And if I take the difference between that and that, this difference vector's going to be orthogonal to everything in l. Just like that. So, this right here would be it's difference vector. That would be x minus the projection of x onto l. And then, of course, this vector right here. This is the one we were defining it. That was the projection onto l of x. Now, what's a different way that we could have written this? We could have written this exact same definition. We could have said it is the vector in l such that-- so we could say, let me write it here in purple. Is the vector v in l such that v-- let me write it this way-- such that x minus v, right? x minus the projection of l is orthogonal is equal to w, which is orthogonal to everything in l. Being orthogonal to l literally means being orthogonal to every vector in l. So I just rewrote it a little bit different, instead of just leaving it as a projection of x onto l. I said hey, that's some vector, v, in l such that x minus v is equal to some other vector, w, which is orthogonal to everything in l. Or we can rewrite that statement right there as x is equal to v plus w. So we can just say that the projection of x onto l is the unique vector v in l, such that x is equal to v plus w, where w is a unique vector-- I mean it is going to be unique vector-- in the orthogonal complement of l. Right? This is got to be orthogonal to everything in l. So that's going to be a member of the orthogonal complement of l. So this definition is actually completely consistent with our new subspace definition. And we could just extend it to arbitrary subspaces, not just lines. Let me help you visualize that. So let's say we're dealing with R3 right here. And I've got some subspace in R3. And let's say that subspace happens to be a plane. I'm going to make it a plane just so that it becomes clear that we don't have to take projections just onto lines. So this is my subspace v right there. Let me draw its orthogonal compliment. Let's say its orthogonal complement looks something like that. Let's say it's a line. And then it goes-- it intersects right there. Then it goes back. And, of course, it would have to intersect at the 0 vector. That's the only place where a subspace and its orthogonal complement overlap. And then it goes behind and you see it again. Obviously you wouldn't be able to you again because this plane would extend in every direction. But you get the idea. So this right here is the orthogonal complement of v, that line. Now, let's have some other arbitrary vector in R3 here. So let's say I have some vector that looks like that. Let's say that that is x. Now our new definition for the projection of x onto v is equal to the unique vector v. This is a vector v. That's a subspace v. The unique vector v, that is a member of v, such that x is equal to v plus w, where w is a unique member of the orthogonal complement of v. This is our new definition. So, if we say x is equal to some member of v and some member of its orthogonal complement-- we can visually understand that here. We could say, OK it's going to be equal to, on v, it'll be equal to that vector to right there. And then on v's orthogonal complement, you add that to it. So, if you were to shift it over, you would get that vector, just like that. This right here is v. That right there is v. And then this is vector that goes up like this, out of the plane, orthogonal to the plane, is w. You could see if you take v plus w, you're going to get x. And you could see that v is the projection onto the subspace capital v-- so this is a vector, v-- is the projection onto the subspace capital V of the vector x. So the analogy to a shadow still holds. If you imagine kind of a light source coming straight down onto our subspace, kind of orthogonal to our subspace, the projection onto our subspace is kind of the shadow of our vector x. Hopefully that help you visualize it a little better. But what we're doing here is we're going to generalize it. Earlier in this video I showed you a line. This is a plane. But we can generalize it to any subspace. This is in R3. We can generalize it to Rn, to R100. And that's really the power of what we're doing here. It's easy to visualize it here, but it's not so easy to visualize it once you get to higher dimensions. And actually, one other thing. Let me show that this new definition is pretty much almost identical to exactly what we did with lines. This is identical to saying that the projection onto the subspace x is equal to some unique vector in V such that x minus the projection onto v of x is orthogonal to every member of V. Because this statement, right here, is saying any vector that's orthogonal to any member of v says that it's a member of the orthogonal complement of v. So that statement could be written as x minus the projection onto v of x is a member of v's orthogonal complement. Or we could call some w. So if you call this your v, and if you call this whole thing your w, you get this exact definition right there. You would have w is equal to x minus v. And then if you add v to both sides, you get w plus v is equal to x. We defined v to be, the orthogonal-- the projection of x onto v. w is a member of our orthogonal complement. And I don't want you to get confused. The vector v is the orthogonal projection of our vector x onto the subspace capital V. I probably should use different letters instead of using a lowercase and a uppercase v. It makes the language a little difficult. But I just wanted to give you another video to give you a visualization of projections onto subspaces other than lines. And to show you that our old definition, with just a projection onto a line which was a linear transformation, is essentially equivalent to this new definition. On the next video, I'll show you that this, for any subspace is, indeed, a linear transformation.