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Unformatted text preview: Page 1 of 2 October 8 th MAT 240 â€“ Linear Algebra Lecture If: ? âŠ‚ , ? = Â¡ , ?Â¢Â¡ ( ? ) = L is linearly independent in V Then: â‰¤ ? & âˆƒ âŠ‚ ? ? . Â£ . = & = ?Â¢Â¡Â¤Â¥? \ Â¦ âˆª Â§ Proof of the above statement: By induction on If = 0 then 0 = â‰¤ ? (as G is a natural number). With = âˆ… the conclusion holds/ Assume lemma is known for any set L` with = Â¨ + 1 = Â© ? 1 â‹¯ ? Â¨ +1 Âª ? = Â© ? 1 â‹¯ ? Â¡ Âª Let ` = Â© ? 1 â‹¯ ? Â¨ Âª , then L` is linearly independent, ` = Â¨ , so by the induction hypothesis lemma holds for L`. This implies ` = Â¨ â‰¤ ? = Â¡ & ` âŠ‚ ? ? . Â£ . ` = ` & ?Â¢Â¡Â¥ ( ? / ` Â¦ âˆª `) = Finally w.l.o.g. ` = Â© ? 1 â‹¯ ? Â¨ Âª So now: ?Â¢Â¡Â¥? Â¨ +1 , â€¦ ? Â¡ , ? 1 , â€¦ ? Â¨ Â¦ = . In particular, ?...
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 Fall '10
 ChristinaSaleh
 Linear Algebra, Algebra, Linear Independence, Vector Space, basis, Linear Algebra Lecture, Assume dim ????

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