601-notes

# 601-notes - MATH 601 AUTUMN 2007 Elementary modifications 1...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

601-notes - MATH 601 AUTUMN 2007 Elementary modifications 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online