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Unformatted text preview: Notes on linear algebra (Monday 17th October. 2016. 23:10) page 168 3.23. Row-echelon matrices Definition 3.144. Let a E ]N and m E N. An a x m-matrix A is said to be zero if and only if A 2 0.1.4.... In other words. an a x m-matrix A is said to be zero if and only if all entries of A are zero. An a X m-matrix A is said to be nonzero if A is not zero. (This does not mean that all entries of A are nonzero!) Definition 3.145. Let m E N. Let a = (131.192.. . ..vm) be a 1 x m-matrix (Le. a row vector of size m). If a is nonzero (i.e.. if not all entries of o are zero). then the smallest i E {1.2. . . .. m} satisfying a. 75 0 is called the pivot index of v and will be denoted by pivind '0. Furthermore. the value of ’0.- for this smallest i is called the pivot entry of v. (Thus. the pivot entry of v is the first nonzero entry of '0.) it as (01.02.03.04w5). then the smallest i E {1.2. 3. 4.5} satisfying 0.- 7E 0 is 2) and pivot entry 6. The row vector (0. 0. —1. —1. 0) has pivot index 3 and pivot entry —1. Example 3.146. The row vector (0. 6. 7. 0. 9) has pivot index 2 (because if we write The row vector (0. 0. O. 0. 0) has no pivot index (since it is zero). Definition 3.147. Let a E ]N and m E ]N Let A be an a x m-matrix (a) Let i E {1.2. .a} be such that row.A is nonzero Then. the cell t__ _'I f ______ 4‘11 t_ _'I1__1|1_ ”1---: IT :__ |1__ - |1_ ____._ _l‘ A _____ __'I_ _ ...
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