00:01
So we have w is a subspace of rn and all the vectors, the p vectors, w form an orthogonal basis.
00:11
And then we have the subspace orthogonal to w in which the q vectors v1 through vq form an orthogonal basis.
00:26
So part a we're asked to explain why all these vectors put together is an orthogonal set.
00:36
Well, so we know that all the ws are orthogonal to each other and the vs are orthogonal to each other.
00:45
So since any wi is in w for all i and any say vj is in the orthogonal subspace to w for all j, this means that that any w dot product with any v has to be equal to zero.
01:19
So therefore the set is orthogonal.
01:25
So for part b, we need to explain why the set in part a spans rn.
01:34
Well, so we have the orthogonal decomposition theorem.
01:38
So for any y in rn, we can write y as a sum of a projection plus some other vector, let's call it z, where the projection is in w, and z is in the complement of w, the orthogonal subspace.
02:12
So since we know this, we can say that we can write y as a linear combination of all the vectors.
02:27
And set a...