00:01
We're given statements about complex vectors, and we're asked to prove these statements.
00:08
So we're told that u, v and w are any n -dimensional complex vectors.
00:18
So vectors in cn.
00:23
In part a, we're asked to show that the dot product distributes on the right across addition.
00:32
In other words, u plus v .w is equal to u .dot -w plus v .w.
00:45
Prove this statement, let's begin on the left -hand side.
00:50
Before we do that, we should define our terms a little more clearly.
00:59
So because u v and w are complex vectors in cn, we'll just say that u has components u1 through un, v has components v1 through vn, and w has components w1 through wn, where all the uis, vis, and wis are complex numbers.
01:29
Well, the left -hand side is u plus v dotted with w.
01:38
First we'll perform the vector addition.
01:41
This gives us u1 plus v1, u2 plus v2, and so on, all the way up to u -n plus vn, dotted with w, which i'll also write as w1, w2, and then performing the dot product for two complex vectors, we get u1 plus v1 times the conjugate of w1 plus u2 2 times the conjugate of w2 and so on, all the way up to un plus vn times the conjugate of wn.
02:31
To get this into the form of the right -hand side, we're going to have to distribute.
02:35
So we use the fact that there is right distribution of complex numbers multiplication over addition.
02:46
So using the distributed property, this is u1w1 conjugate plus v1w1 conjugate plus u2 w2 conjugate plus v2 w2 conjugate and so on, all the way up to un times wn conjugate plus vn times wn conjugate.
03:07
And at this point, i'm going to regroup terms using commutative property of addition.
03:15
And so i get u1 times w1 conjugate plus u2 times w2 conjugate all the way up to un times wn conjugate, plus v1 times w1 conjugate, plus v2 times w2 conjugate, all the way up to vn times w n conjugate and this is clearly equal to the complex vector u dotted with the complex vector w plus the complex vector v dotted with w and this is the right -hand side of the equality and so we've proven this equality in part b we're now asked to prove the same statement except for we're distributing the dot product from the left so we're proving left distributivity of the dot product over addition.
04:16
While it seems like we're going to have to do the same steps as part a, there's actually a faster method.
04:22
Beginning with the left hand side, we have w dotted with u plus b...