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Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 162 0 Third, choose a value for w (3) There are a — 2 possible choices for this (since as (3) has to belong to {1, 2,. . . , a}, but must not equal any of w (1) and w (2), since we want the 14 numbers to (1) , w (2) , . . ., w (a) to be distinct). :- Fourth, choose a value for w (4). There are a — 3 possible choices for this (since It} (4) has to belong to {1, 2,. . ., a}, but must not equal any of w (1) , w (2) , w (3), since we want the 11 numbers w (1) ,w (2) , . . ., w (n) to be distinct). 1. And so on. 0 At the last setep, choose a value for w (11). There are a — (a — 1) possible choices for this (since as (a) has to belong to {1,2,. . . , a}, but must not equal anyofw (1),w (2),...,w (n — 1), sincewewanttlneanumbersw(1),wI (2),. ..,w(a) to be distinct). We thus get a total of n - (a — 1) - (n — 2) . (a — 3) ----- (n — (a — 1)) possible choices. In other words, we get a total of a! possible choicesmfl. Hence, there are precisely at! permutations of {1, 2,. . . , a}. As we have said, this proves Proposition 3.114. D Here is a further property of permutation matrices constructed out of permuta- tions: ...
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