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Home » College Playground » Calculus - Real Analysis » Calculus Computations and Tutorials

linerly independent
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mayhem
Poincare Conjecture

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linerly independent

Show that, functions $e^{k_1t},e^{k_2t},e^{k_3t},...,e^{k_nt}$ are linerly independent. for $i \neq j , k_i \neq k_j$

Posted: Tue Nov 03, 2009 2:44 pm

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jmerry
Birch & Swinnerton Dyer

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Build the Wronskian. It's essentially a Vandermonde determinant.

Posted: Tue Nov 03, 2009 4:01 pm

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WWW
Poincare Conjecture

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Re: linerly independent

mayhem wrote:

Show that, functions $e^{k_1t},e^{k_2t},e^{k_3t},...,e^{k_nt}$ are linerly independent. for $i \neq j , k_i \neq k_j$

Suppose $\sum_{m=1}^{n} c_me^{k_mt}=0$ for t in some interval. Then $e^{k_1t}(c_1+\sum_{m=2}^{n} c_me^{(k_m-k_1)t}) = 0$ and therefore the term in parentheses equals $0$ . So differentiate that term and use the induction hypothesis (which I never stated, but you get the idea).

Posted: Tue Nov 03, 2009 9:19 pm

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