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001_601-elementary-modifications

001_601-elementary-modifications - MATH 601 AUTUMN 2008...

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MATH 601, AUTUMN 2008 Elementary modifications 1. Definitions. Let X be a finite system of vectors in a vector space V over a field K . Another such system X is said to arise from X by an elementary modification if X is obtained from X by replacing some entry v with (a) a multiple cv for some scalar c = 0, or (b) v + aw , where w occurs in X at a different place than the original entry v and a is a scalar, and leaving X otherwise unchanged. Note that, in this case, if X is linearly independent, then so must be X . Also, it follows that X then arises from X by an elementary modification as well. If, instead of (unordered) systems discussed above, we deal with finite systems that are ordered , we allow a third kind of elementary modifications, in which X arises from X by switching two neighboring entries, and leaving all remaining entries unchanged. Given two finite systems X, X (ordered or not) in a vector space V , we will say that X and X are equivalent , if X is obtained from X as a result of a finite number of successive elementary modifications. (The number of these modification may be 0, so that
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