Could you please walk me through this conditioning, I cant seem to get my hear around this type of thing, how do you come to the conclusion that they are independent etc?

Let X~N(1,0) V~Xk2

define T = X / (V/k)

So, T | V { V = v} = X (k/v) | {V = v} = X (k/v) ~ N(0, k/v)