combinepdf-min-0286.pdf

# combinepdf-min-0286.pdf - Notes on linear algebra(Monday...

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Unformatted text preview: Notes on linear algebra (Monday 17th October. 2016, 23:10) page 133 3.18. (*) Some proofs about the swapping matrices Proof of Prapasition 3.105. Clearly, 1"“va is an n x m-rnatrix. Let x and y be two elements of {1,2,. . . ,a}. Proposition 3.44 (applied to a, m, x and 3; instead of m, p, a and a) shows that ExfyC is the a X m-matrix whose x-th row is the y-th row of C, and whose all other rows are filled with zeroes. Thus, rowI (ExfyC) = rowy C (110) (since the x-the row of ExfyC is the y-th row of C) and rowi (EWC) = 013).”?I for every 1' E {1.2, . . . ,a} satisfying 1' 71 x (111) (since all other rows of ExfyC are filled with zeroes). Now, forget that we ﬁxed 3!: and y. We thus have proven (110) and (111) for every two elements 3: and y of {1, 2, . . ., a}. In particular, every y E {1, 2,. . .. a} satisfies (by (111), applied to i = a and x = 0 (since a 7E 32)) and rowv (EWC) : 01m (113) “an {1111 nnnliar'i 1-an — a1 and ’1" — 1: {Qiﬁr‘ﬁ at —;L 1:“ ...
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