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Lecture 6

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Kathleen Smith

w 2 w3 w1 w 2 w3 2w 1 e 3 w 2 w3 w1 n Definition: Let {w , w ,1▯▯▯ 2w } be a sek of vectors in R . A dependence relation in this set is an equation of the form λ1w +1λ w +2▯▯▯2+ λ w = 0 k k with at least one λ ▯i 0. If such a dependence relation exists, then {w , w ,1▯▯▯ 2w } is a k linearly dependent set of vectors. Otherwise, the set of vectors is linearly independent. n Theorem: LetW be asubspaceof R . A subset{w , w , ▯▯▯ ,w }1 2 k of W is a basis for W if and only if the following conditions hold. (i) The vectors w , w , ▯▯▯ ,w span W. 1 2 k (ii) The vectors w , 1 , ▯2▯ ,w are likearly independent. To find a basis for W = sp(w , w1, ▯2▯ ,w ) we pkt ▯ ▯ A = w w 1 2 ▯▯▯ w k and row reduce to B where B is in echelon form. Then the set of w sjch that the j thcolumn of B contains a pivot is a basis for W. A set of vectors {v 1v ,2▯▯ ,v }kis independent if and only if the ▯ ▯ echelon form of A = v v 1▯▯2v k has a pivot in each column. Theorem: Let W = sp(w ,w ,1▯▯▯ 2w ) ⊂ R , k ≥ k. Ifn v 1v 2▯▯▯ ,v ame independent vectors in W, then m ≤ k. Corollary: Invariance of Dimension Any two bases of a subspace W con
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