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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 171 (c) Proposition 3.151 (a) shows that the first k rows of A are nonzero. In other words, row,“ A is nonzero for each i E {0, 1,. . . ,k — 1}. The condition in Defi- nition 3.149 thus shows that each i 6 {1,2, . . . ,lc — 1} satisfies pivind (row, A) < pivind (row,+1 A) (since A is a row echelon matrix). In other words, pivind(row1 A) < pivind (row; A) < - - - < pivind (rowk A) . (144) This proves Proposition 3.151 (c). (13) Proposition 3.151 (a) shows that the first 16 rows of A are nonzero whereas the last a — k rows of A are zero. Thus, the pivot cells of A are the pivot cells in the first It rows of A. In other words, they are the k cells (1, pivind (row1 A)) , (2, pivind (rowz A)) , . . ., (k, pivind (row,C A)) . This proves Proposition 3.151 (b). D Proposition 3.152. Let A be a matrix. (a) Each row of A has at most one pivot cell. (b) Assume that A is a row-echelon matrix. Each column of A has at most one pivot cell. Proof of Proposition 3.152. (a) This is clear from the definition of a pivot cell. (More precisely, each nonzero row of A has exactly one pivot cell, whereas a nonzero row ...
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