00:01
We're going to find the orthogonal projection of vector v equals 0 -0 -0 -negative 6 onto the subspace v -r -4 span with the vectors v1 equal 1 -81 -1 -1 -1 -2 equal 1 -8 -1 -1 -1 -1.
00:23
So first thing we note is that the set of vectors v1, v2, b3 which spanned the subspeighted, space fee of r4 are orthogonal to by two different vectors are orthogonal to each other so is an orthogonal set in r4 or a subset of our r4 that is if we calculate the inner part of v1 and v2 for example we get multiplied together the corresponding components and add up those products so we get 1 times 1 plus negative 1 times negative 1 plus negative 1 times 1 plus negative 1 plus negative 1 times 1 the result is 1 1 again negative 1 and negative 1 so 1 negative 1 cancel out and 1 negative 1 cancel out we get 0 if we do v1 and v3 in a product we get 1 times 1 plus negative 1 times 1 plus negative 1 times 1.
02:02
And plus negative 1 times 1.
02:06
That is 1 negative 1, 1, 1 minus 1 0, 1 0.
02:17
So we get 0.
02:19
And finally, you see we have combined b1, b2, b1, 3, and now we can combine v2, 3.
02:25
So we have proved all the possible inner products of two different vectors out of the set of three vectors v2 v3.
02:33
And we get b2b3 that says 1 times 1 plus negative 1 times 1 plus 1 times 1, plus 1 times 1.
02:58
That is 1, negative 1, negative 1, 1, 1, and you see 1 minus 1 is 0, negative 1 is 0.
03:11
So we get 0.
03:13
So we have verified that the set v1 v2, b2, 3rd, sootum.
03:18
Then when the set of vectors that span the subspace onto which we want to project a given vector, where then we have a simplified formula given like this.
03:34
So the projection onto v of vector v is equal to the inner product of vector v.
03:46
V, vector v1 in the set over inner product of v1 by itself times v1 plus inner product of vector v2 over v2 inner product with itself.
04:02
And that scalar times v2 plus the inner part of v times v3 divided by the inner part of v3 by itself and all that scalar times v3.
04:15
That is the formula we use to project a vector onto a unique vector, a unique non -zero vector, is then add together applied for each of the vectors that span the subspace onto which we want to project the given vector.
04:33
But that is only true, that formula can be used only in the case when the set of vectors is finding the subspace are orthogonal, which is the case right here.
04:45
Okay, so we got to calculate these inner products.
04:50
And you see we have to calculate in the numerators, the inner products of vector v with each of the vectors v1, v2, b3.
04:58
But you see vector v here has all zero except for the last component, negative 6.
05:04
So we'll multiply the corresponding components of v, which is of the components of each of these three vectors.
05:10
The only term that survived, let's say, that is not zero.
05:15
Maybe not zero is negative 6 times the fourth component of each of the vectors.
05:21
V1 v2, be 3...