w 2
w3 w1
w 2
w3
2w 1
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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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