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Unformatted text preview: MATH 601, AUTUMN 2007 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 6 = 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 each system is equivalent to itself.) Obviously, if X and X are equivalent, then # X = # X , Span X = Span X , that is, they have the same number of elements, and span the same subspace. In particular, if one of X, X spans V , then so does the other. 2. The result. Proposition. Let X, Y be finite unordered systems of vectors in a vector space V over any scalar field K , such that X is linearly independent, and Y spans V...
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