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Unformatted text preview: Page 1 of 2 October 8 th MAT 240 Linear Algebra Lecture If: ? , ? = , ? ( ? ) = L is linearly independent in V Then: ? & ? ? . . = & = ?? \ Proof of the above statement: By induction on If = 0 then 0 = ? (as G is a natural number). With = the conclusion holds/ Assume lemma is known for any set L` with = + 1 = ? 1 ? +1 ? = ? 1 ? Let ` = ? 1 ? , then L` is linearly independent, ` = , so by the induction hypothesis lemma holds for L`. This implies ` = ? = & ` ? ? . . ` = ` & ? ( ? / ` `) = Finally w.l.o.g. ` = ? 1 ? So now: ?? +1 , ? , ? 1 , ? = . In particular, ?...
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