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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 (nontrivial l.c. of distinct elements of ° ∪ ²´³ → ° ∪ ²´³ is linearly dependent....
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 Fall '10
 ChristinaSaleh
 Linear Algebra, Algebra, Vector Space, basis, abstract linear algebra, Abstract Linear Algebra Lecture

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