18.06 Linear Algebra, Spring 2010
Transcript – Lecture 16
OK. Here's lecture sixteen and if you remember I ended up the last lecture with this
formula for what I called a projection matrix.
And maybe I could just recap for a minute what is that magic formula doing? For
example, it's supposed to be -- it's supposed to produce a projection, if I multiply by
a b, so I take P times b, I'm supposed to project that vector b to the nearest point in
the column space. OK.
Can I just -- one way to recap is to take the two extreme cases. Suppose a vector b
is in the column space? Then what do I get when I apply the projection P? So I'm
projecting into the column space but I'm starting with a vector in this case that's
already in the column space, so of course when I project it I get B again, right.
And I want to show you how that comes out of this formula.
Let me do the other extreme. Suppose that vector is perpendicular to the column
space.
So imagine this column space as a plane and imagine b as sticking straight up
perpendicular to it.
What's the nearest point in the column space to b in that case? So what's the
projection onto the plane, the nearest point in the plane, if the vector b that I'm
looking at is -- got no component in the column space, it's sticking completely --
ninety degrees with it, then Pb should be zero, right. So those are the two extreme
cases. The average vector has a component P in the column space and a component
perpendicular to it, and what the projection does is it kills this part and it preserves
this part. OK.
Can we just see why that's true? Just -- that formula ought to work.
So let me start with this one. What vectors are in the -- are perpendicular to the
column space? How do I see that I really get zero? I have to think, what does it
mean for a vector b to be perpendicular to the column space? So if it's perpendicular
to all the columns, then it's in some other space. We've got our four spaces so the
reason I do this is it's perfectly using what we know about our four spaces. What
vectors are perpendicular to the column space? Those are the guys in the null space
of A transpose, right? That's the first section of this chapter, that's the key geometry
of these spaces.
If I'm perpendicular to the column space, I'm in the null space of A transpose.
OK. So if I'm in the null space of A transpose, and I multiply this big formula times b,
so now I'm getting Pb, this is now the projection, Pb, do you see that I get zero? Of
course I get zero.