Linear indepedent functions and complex conjugation

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Discussion Overview

The discussion centers on the linear independence of sets of functions, specifically examining whether the complex conjugates of an infinite set of linearly independent functions maintain linear independence. Participants also explore the implications of combining the original functions with their complex conjugates.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant posits that if the set \{ \varphi_i \} is linearly independent (LI), then the set \{ \varphi_i^\ast \} must also be LI, suggesting that a violation would contradict the independence of \{ \varphi_i \}.
  • The same participant expresses uncertainty regarding the claim that the union of \{ \varphi_i \} and \{ \varphi_i^\ast \} is LI, acknowledging the possibility of counterexamples.
  • Another participant introduces a specific example, proposing the set \{i f, f\} where f is a real-valued function, to challenge the previous claims.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the linear independence of the union of the sets or the independence of the complex conjugates. Multiple competing views and uncertainties remain in the discussion.

Contextual Notes

The discussion highlights potential limitations in the assumptions regarding the properties of linear independence in relation to complex conjugation, as well as the need for specific examples to clarify the claims made.

jdstokes
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Suppose [itex]\{\varphi_i\}[/itex] is an infinite set of linearly independent functions. Is [itex]\{ \varphi_i^\ast \}[/itex] linearly indepedent? How about [itex]\{ \varphi_i \} \cup \{ \varphi_i^\ast\}[/itex]?
 
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Well, what are your thoughts on the matter?
 
Well, if [itex]\{ \varphi_i \}[/itex] is LI then [itex]\{ \varphi_i^\ast \}[/itex] is also trivially LI, because if it wasn't then you could just take the complex conjugate violating linear indepedence of [itex]\{ \varphi_i \}[/itex].

It's not clear if my second claim is true, although I'd like it to be, I suspect there are counterexamples waiting to be found. I'd like to be proven wrong, however.
 
What if we take, say, the set [itex]\{i f, f\}[/itex], where f is some real-valued function?
 

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