00:01
In this question, we have to show that the ring m which is defined as the collection of the matrix of the form ab minus ba such that ab belongs to the integer, this ring is isomorphic to the ring of gaussian integers.
00:23
So, to show this isomorphism, we are going to define a map from this m to the ring of gaussian integer as f of a equals to a plus b times of i where ab belongs to z.
00:43
First of all, our aim is to show that f is a ring homomorphism, for that we first consider f of a plus b, this will be f of, here a belongs to m, so f of a, i will take as a1 b1 minus b1 a1 plus b, let's say it is a2 b2 minus b2 and a2, so this comes out to be f of a1 plus a2 b1 plus b2 minus b1 plus b2 a1 plus a2, correct? so, this comes out to be according to the definition it is a plus bi, so i will get a1 plus a2 plus b, in place of b we have b1 plus b2 times i, this we can split as a1 plus b1 i plus a2 plus b2 i, so this is equals to f of a1 b1 minus b1 a1 plus this is f of a2 b2 minus b2 a2, so the first condition is satisfied and this is true for every a and b in the set m.
02:20
Next condition is to show that f of ab is f of a times f of b, so f of ab if we consider, so this comes out to be the multiplication i am directly writing, it comes out a1 a2 minus b1 b2, here we get a1 b2 plus a2 b1 minus b1 a2 plus a1 b2 and this is minus b1 b2 plus a1 a2, this is the multiplication ab...