00:03
So we fix v in v and we want to find p and s and since p is an s, this means an s has a basis, this means p can be written as c -i -u -i for for i equals 1 to n and c -i in reels.
00:27
And what's the definition of the projection? the projection is the p such that you're minimizing v minus p squared.
00:37
So you're trying to minimize this for p and s.
00:42
So this would be what? well we can write this as minimizing over c -i of v minus sum from i equals 1 to n, c -i -u -i.
01:01
Okay, now let's write out with this.
01:04
So we can write this as a function, so this is a basically a function of c1, c -n, c -n, this is a function of n real variables and it gives you a real number.
01:23
So this is basically a usual multivariable calculus problem.
01:29
Now let's calculate what this f of c1cn is.
01:36
So using inner products, this norm can be written as norm of v squared plus norm of this ciui squared and then you get a minus 2 times.
01:52
Sum of c i v u i using linearity okay so now let's further try to simplify this the main thing being we want to know what some what this norm is well using inner products this is the sum of i equals one to n of c i ui and then it's using another index j equals 1 to n of c .j.
02:31
U .j.
02:34
So using linearity, you can bring this from both sums out.
02:42
N of c .i.
02:44
C .j.
02:46
U .i.
02:47
U .j.
02:50
But you know the uis are orthanormal basis.
02:54
So this is equal to zero if i and j are different.
02:59
And it's equal to 1 if i is equal to j.
03:04
The only so basically what you get is that this is equal to sum from i equals one to n of c i squared okay so in total you get that f of all these c1 c2 all the way to cn is equal to the sum from i equals one to n c i squared minus two uh times sum of c i uh v u i and then plus norm of v squared...