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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 78 Case 2: We have 1' 75 v. Case 3: We have neither i 7E a nor j 75 '0. (In fact, it is possible that we are in Case 1 and Case 2 simultaneously. But this does not invalidate our proof; it is perfectly fine if the cases ”overlap”, as long as every possible situation is covered by at least one case.) We shall prove (63) in each of the three cases: 1. Let us first consider Case 1. In this case, we have 1' 7E a. Hence, (1' j) 7E (a 0). Thus, 5(ia).{u.v) = 0. Comparing this with of!“ 5w = 05m = 0 we find (singggéa) ash-MW) = afruaj’v. Hence, (63) is proven in Case 1. 2. Let us next consider Case 2 In this case we have j 7é o. Hence e, (i j)7é (a 0). Thus 5( 1;, MW) — —.0 Comparing this with at” 5;“,2: — —§WO 2 0w efind —0 (sincEjav) ‘SWMHM = awaijffl. Hence, (63) is proven in Case 2. 3. Let us finally consider Case 3. In this case, we have neither 1" 7E a and j 7E ’0. Hanna 3" = 11 (oinr‘p nnt i :é “‘1 nnrl i = 7': (nine:- nnt 1' :é to] A: n Fnhqpnnphr‘p ...
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