Visualizing a projection onto a plane Visualizing a projection onto a plane. Showing that the old and new definitions of projections aren't that different.
Visualizing a projection onto a plane
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- 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.
- 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
- 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.
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