2. Do one more step in the proof of Proposition 1 by showing (L.(i),L.(k)) = 0. q 9 () Proposition 1: If qE83, then th
Posted: Tue Sep 07, 2021 7:34 am
- 2 Do One More Step In The Proof Of Proposition 1 By Showing L I L K 0 Q 9 Proposition 1 If Qe83 Then Th 1 (24.75 KiB) Viewed 79 times
Proposition 1: If qE83, then the "left translation" La : HH given by L (9') = qa' 4 R is an orthogonal map of to 4 Proof: As vector spaces over R, H and are the same. So La is surely a linear map of R for if a,be R and a, B E H we have L (aa + b) q(a + b) = aqa + b 28 = aly(«) + bl. bL (8) 9 To see that La is orthogonal, it suffices to show that preserves the perpendicularity (using , for R) of the four unit vectors 1,1,j,k. For example, let q = a + ib + jc + kd and 61 We get for £4). calculate (L,(i), (j)) (using !,) ad + bc - bc - da = 0. For (L (1),L, (1) we get (a + ib + jc + kd, ai - b - kc + jd) = -ab + ab + dc - dc = 0.