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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 128 (applied to b 2 col),- (In) and b, = 5,3), and r = j) says that there exists a column vector v = (191,222, . . . ,vn)T of size a satisfying At: 2 col; (In) and 01 = 7.22 = ... = 1 vj_1 = 0 and “a, = A—aj’j' Denote this vector v by am, and denote its entries a :21, '02, . . . , ”0,, by UL}, "020,-“ . . , on”, (in order to stress that they depend on j). Now, forget that we fixed j. Thus, for each j E {1,2,...,r1}, we have found a column vector Um = (0101*, “02”,“, . . ., fluff) T (104) of size a satisfying Avg] 2 col),- (In) (105) and 01”,“ = 1120,‘ = - - - = UJ,‘_1J = 0 (106) and 1 UN 2 Edy-J. (107) Altogether, these are a column vectors 7.9m, am. . . ,UM of size a. We can assemble them into a matrix B: Namely, set B = (”aha-5n, lizjgn' ...
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