01.2.5 Unit ball Part 2

01.2.5 Unit ball Part 2


Okay so let’s see what these unit balls
look like. Let’s start by focusing on the 2-norm. So the set where the 2-norm
is equal to 1 is of course equivalent to the set where chi_0 squared plus chi_1
squared is equal to 1. And that’s something that almost surely you have
plotted before. That’s the unit circle, the circle with radius 1 centered at 0.
So that particular set is this set right here. Alright! So now let’s look at the set
of all vectors with 1-norm equal to one, in other words, [all points distance
one by the, well] all points such that the absolute value of the first coordinate plus
this absolute value of the second coordinate is equal to one. And let’s
think this through. Well, it’s pretty obvious that all of these points are
going to lie between minus 1 and 1 for the chi_0 component and between minus 1 and 1 for chi_1. So it’s going to be somewhere there. What are some points that are
obviously on this set? Well, if chi_0 is equal to 1, then chi_1
has to be 0. So this point has to be on there. And if chi_0 is equal to 0 then
chi_1 can be either 1 or -1. And of course, by the same reasoning we see that the point -1, 0 is also going to be in that set. Alright? Now, absolute value
always a little bit difficult to deal with. Well, if we know that chi_0 and chi_1 are positive, in other words, if we ask the question what point in this quadrant
are in this set, then this becomes the set chi_0 plus chi_1 is equal to 1,
which we recognize as a line. And that line has to go through these two points.
So therefore we know that in this quadrant, all of those points are in the
set. Similarly, you can reason through the fact that all of these points are in the
set. All these points are in the set. And, all of those points in the set. And as a
matter of fact, only those points are in the set. So now we have a picture of all
points to which vectors with 1-norm equal to one point.
Alright? Alright, so let’s do the last one. That one is a little bit harder. Alright,
let’s come up with some points that would be on there. Let’s see. (1,0)
would be in this set. (0,1) would be in that set. (-1,0) would be in
that set. [ (-1,-1), sorry, -1] (0,-1) would be in
that set. So far so good. But (1,1) is also in that set. So is
(-1,1). So is (-1,-1). So is (1,-1). And if you kind of
reason through it that way, you find out that all points such that
chi_0 is equal to 1 and chi_1 is between -1 and 1 [are on that] are in the set.
And so are all of these points, and these points. So there is the unit ball or an
infinity-norm.

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