00:01
Okay, we got a basis consisting of these four matrices.
00:04
And you have an inner product.
00:07
If you have the inner product of u and b, where you and b are matrices is the trace of e transpose v.
00:25
So we have to use graham schmidt orthogonalization to construct an orthogonal basis.
00:34
So we are going to select v1, our first vector.
00:44
We can call a matrix a vector, a set of all matrices with scalar multiplication and that matrix addition make up a vector space.
00:54
So i'm going to say vector a lot when i mean matrix in this example.
00:59
So i will select vector one to be the first matrix in the list, 1 -1 -0.
01:04
Then vector two has to be the second vector in the list, one zero, one, one.
01:19
So the second matrix in the list, one zero one.
01:23
But then we subtract one zero one interproduct of that with the inner part of the first vector in our local basis.
01:43
We divide that by the inner product with 11101110 and then we multiply that by the first vector in the list.
02:10
So one one zero excuse me, that should be one one one zero.
02:29
All right, so we need the trace of u transpose v so we'll do this here.
02:49
11011110 this inner product is the first guy transposed so that's going to be one zero one one times and it's the trace of this one i'm about to write times um one one one zero all right so we're going to do the trace of this um um um 1 times 1 so this is going to be a 2 1 times 1 plus 1 times 1 is 2 1 times 1 plus 1 times 0 is 1 plus 1 times 1 times 1 is 1 0 so the trace of this is 2 now let's do inner product let's do this guy right here the inner product of 111110, 1 ,110.
04:10
Oh, but i should have mentioned, the trace is just you sum the diagonal entries.
04:26
And so now we do the trace of the first guy transposed.
04:36
So 1 ,110, and 1 ,110.
04:52
So it's equal to it self -transpose.
04:55
So do the matrix multiplication.
05:01
We get 1 times 1, 1 times 1 is 2.
05:06
1 times 1 times 0, 0 0 0 times 0 is 1 times 1 is 1 times 1 times 1 0 is 1 times 1 0 is 1 0 0.
05:14
3.
05:21
So putting it all together, we get our second vector in the list is 1110 minus ms step.
05:46
That's one zero one one one zero one minus two thirds times one one one one zero this is going to equal one minus two thirds zero minus two thirds um one minus two thirds and one minus two thirds and one minus zero which is going to give me three thirds, one -third, negative two -thirds, one -third, and one.
07:04
This is a b -2.
07:06
So now our basis looks like this.
07:10
We have a vector.
07:12
The first vector is 1 -1 -0.
07:20
The second vector is, i just wrote it.
07:26
Let's multiply that.
07:28
So here's something that will make our lives easier.
07:32
It doesn't say you need an orthonormal basis.
07:35
It says we just need an orthogonal basis.
07:47
So i'm going to multiply this matrix by a 3.
07:55
Get 1, negative 2, 1, 3.
08:04
Let's check to see if it's orthogonal.
08:08
So if i do the trace of the first guy, transpose, so 1 -1 -0 times 1 -9 -2.
08:21
One three i get one times one is one plus one times one is two one times negative two is negative two one times three so negative two plus three is one one times one is one is one zero that's one then negative two right here we'll see that the trace of that is zero so they're in orthogonal so that satisfies the condition of checking along the way to see if the vectors you're getting are orthogonal so right now our basis contains these two vectors to get the third vector in the basis v3 you take the third basis vector in our list which was 1101 and you subtract 1101 the inner product of 1101 with the first vector in your basis, 1110, and then you divide it by the inner product of the first vector in the basis with itself.
10:08
So 1 .1 .10, 1110.
10:15
And then you multiply that by 1110.
10:29
Then you do the same thing with the second vector in your list.
10:35
So you take 1101, which is this guy right here.
10:50
And then we take this vector, 1 -2 -13, divided by 1 -2 -13, and the product itself, 1 -2 -13...