combinepdf-min-0323.pdf

# combinepdf-min-0323.pdf - Notes on linear algebra(Monday...

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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 170 o a nonzero row of A cannot follow a zero row of A, and o the first nonzero entry in each nonzero row of A is strictly further right than that in the row before. Here are some ﬁrst simple observations: Proposition 3.151. Let A be an a x m-matrix. Assume that A is a row-echelon matrix. Let k be the number of nonzero rows of A. (a) The ﬁrst k rows of A are nonzero, whereas the last a — k rows of A are zero. (b) The pivot cells of A are the k cells (1, pivind (row1 A)) , (2, pivind (rowz A)) , . . . , (k, pivind (row,C A)) . (c) We have pivind (rowl A) < pivind (rowz A) < - - - < pivind (rowk A) . Proof of Proposition 3.151. The matrix A is a row-echelon matrix. Thus, the condi- tion in Deﬁnition 3.149 shows that, for each i 6 {1,2, . . .,a m 1}, the following holds: if raw.- . 1 A is. nnnzern- then row; A is. nnnzprn. {143) ...
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