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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 79 for all i E {1,2,...,a} and j 6 {1,2,. . .,m} (according to Proposition 3.48). Fur- thermore, for anyi 6 {1,2,. . .,a} andj E {1,2,...,p}, we have m by Proposition 2.28, (E“ri’c)ia _ E (Bash, Chi ( applied to A = Em: and B = C ) =5i,u5k,v (by (64), applied to it instead of j) m m = Z {Sirucskflckj = Z (Siackjakm = 5i,aco,j (65) 36:1 k=1 (by Proposition 2.40, applied to 1, m, o and (Stacka' instead of p, q, r and ak). Now, we can prove both Claims easily: Proof of Claim 1: For every j E {1, 2,. . . , p}, we have (EWChJ = 51,!“ CU“,- (by (65), applied to i : a) (singima) : CUJ‘ In other words, for every j 6 {1,2, . . ., p}, the (a, j)-—th entry of the matrix ERIUC equals the (o, j)-th entry of the matrix C. In other words, each entry of the a-th row of 13.. ..C eouals the eorresaoondinar entrv of the U-th row of C. In other words. the ...
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