00:01
For this problem, we are given that t star times t is equal to zero, and we're asked to prove that that implies that t equals zero.
00:08
Okay, so there's more than one way to prove this, but here's one such way.
00:12
So first, i'm going to let v1 through vn be the columns of t.
00:18
And in other words, the matrix t is equal to this matrix.
00:24
Okay, then this implies that t star times t is equal to the product of these matrices.
00:40
So first of all, t star, it will look like this.
00:47
So the first row of t star will be v1 star, and recall that the star is the conjugate transpose.
00:54
So v1 star is a row vector.
00:57
Similarly, v2 star, this is also a row vector, dot, dot, dot, until we get to vn star.
01:10
And then we're going to multiply this by t.
01:17
Okay, so recall that in matrix multiplication, you're multiplying.
01:22
So for this, let's say for this first entry right here, right, to find it, you go across the first row and then down the first column, right? so this first entry, this will be v1 star and then times v1, right? then for the next entry, we have v2 star times v1...