ALA240-2009 - October 1st

# ALA240-2009 - October 1st - ´ ∈ µ is independent unless...

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October 1 st , 2009 MAT240; Abstract Linear Algebra Lecture Goal: Every v.s. has a basis. So while we don’t have to work in coordinates, we always can. °±?? °² = ³´ ? µ ? µ ? µ ∈ ¶ , ? µ ∈ ° ? µ =1 · Proposition: 1. Span(S) is a subspace of V. 7 ´ ? µ ? µ = ´ (7 ? µ ) ? µ ? µ = ¸? µ µ ≤ ? , ? µ−? µ > = ´ º µ ? µ 2. If ° 1 ⊂ °±?? ( ° 2 ) , then °±?? ° 1 ² ⊂ °±?? ( ° 2 ) °±?? ° 2 ² = ? ⊂ » → ° 1 ⊂ ? → °±?? ° 1 ² ⊂ ? = °±?? ( ° 2 ) Definition: A subset ° ⊂ » is called (“wasteful”) linearly independent if ¼º?½?¾¼²? µ & ¿µ¼Àµ?ºÀ ? µ ∈ ° s.t.: ? 1 ? 1 + ⋯ ? ? ? ? = Á ? µ ? µ = 0 and not all ? µ ’s are 0. Definition: ° ⊂ » is linearly independent if it is not linearly dependent. ↔ Á ? µ ? µ = 0 → ∀µ ? µ = 0, ? µ ∈ ¶ ??¿ ? µ ∈ ° Example: In Â Ã?? = » ° = ÄÅ 1 0 0 1 Æ Å 0 1 0 0 Æ Å 0 0 1 0 Æ Å 0 0 0 1 ÆÇ is linearly independent

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? 1 0 0 1 ± + ? 0 1 0 0 ± + ? 0 0 1 0 ± + ? 0 0 0 1 ± = 0 0 0 0 ± ? ? ? ? ± = 0 0 0 0 ± → ? = 0, ? = 0, ? = 0, ? = 0 Comments: 1. 0 = ² ³ is linearly independent 2. ²´³ ;
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Unformatted text preview: ´ ∈ µ is independent unless u = 0, but if u = 0 is dependent. 3. Suppose ° 1 ⊂ ° 2 ⊂ µ a. If ° 1 is linearly dependent, ° 2 is linearly dependent. b. If ° 1 is linearly independent, know not about ° 2 . c. If ° 2 is linearly independent, so is ° 1 . d. If ° 2 is linearly dependent, know not about ° 1 . 4. If S is linearly independent and ´ ∈ µ \ ° then ° ∪ ²´³ is linearly dependent iff u is a linear combination of elements of S. ↔ ´ ∈ °¶?· ( ° ) Proof: Suppose ´ ∈ °¶?·¸°¹ → ´ = º ? » ´ » ? » ∈ ¼ , ´ » ∈ ° ( ?»½¾»·?¾ ) ↔ 1 ´ − º ? » ´ » = 0 (non-trivial l.c. of distinct elements of ° ∪ ²´³ → ° ∪ ²´³ is linearly dependent....
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ALA240-2009 - October 1st - ´ ∈ µ is independent unless...

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