Question
PS) Prove the following relationships:(10 points)av aT8s aPaT dPav 8S
PS) Prove the following relationships: (10 points) av aT 8s aP aT dP av 8S
Answers
Using the relation $1 / v_{g}=d \kappa / d \nu,$ prove that $$\frac{1}{v_{g}}=\frac{1}{v}-\frac{v}{v^{2}} \frac{d v}{d v}$$
Okay, So in this question, we want to show that the projection operator on the union offset is going to be the projection operators on the individual sets union does even individual sets. So first night that I just write it in a different notation. So they wrote, I want I to I am here. I swear I am I because it just seemed fives arriving. But it's essentially the same thing. So first, let's do it's right what this left hand side means to the projection operators off I That's in the set. I off our U. S. Is going to be equal to face of by one ace Up by two. Boba, Ace of I am on the condition That's a one a two Bob a la A and is in a meeting s so similarly the projection off by in i o He's then going to be a sub by one ace of by tube Baba Blah, Ace of I am on the condition a one a two of our a n He's in AA and P I in I off s don't be equal to a Some are one a certified to Allah blah ace of I am on the condition. A one a two a in this. So to union Theseus have together he I in Iove our union with P I in iron Off s. This is going through the exact same thing. He Aye, aye. One a guy to While I am on the condition that this is true after a one a two a what A. M is in our or this whole thing isn't s now, Ryan, or is equivalent to writing our meaning? Yes, and I'm not going to write the whole thing again because I was sick too long. But essentially this Remove this and this becomes our U. S. And you see that these true lines are equal to each other and so you don't
Okay, so we want to show that in general, the projection operators all eye on our set minus s is not in general equal to P I in Irish. Here I come. Oh, are set minus p I in Iove set best. So the enemy dunny you only need for that this is not equal in general, if you can come up with a counter example So the counter example that they give is going to be aren't that is equal to a beast? Just one people and s being the set A C Then what happens is offset. Minus s is just going to be the set A B answer. We take a projection operators to B P one. So then the projection operators Paul Ah, Is this going to be a so the set? Okay. On the other hand, if we do this first, the projection operators all our set minus projected operating off s within the production off the projection off. Brake on artist returns us, eh? The projection operator operating on s is us, eh? He said months. Hey, these who says are exactly the same. So you're left with an empty set. So in one case. This is also almost in one case, the left of the set off letter A. In this case, he left with an empty set, and so they're not equal in general.