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## Homework Statement

If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction?

I thought for sure yes, but the way I particular question I just worked through went seemed to suggest it shouldn't be obvious, so perhaps not always guaranteed too.

## Homework Equations

So I have ## \hat{P} F(x) = 0 F(x) ##

##\hat{P}G(x)=16F(x)+G(x)##

## The Attempt at a Solution

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##=> \hat{P}(16F(x)+G(x))= \hat{P}(16F(x))+\hat{P}(G(x))=0+\hat{P}(16F(x)+G(x))## therefore ##16F(x)+G(x)## is an eigenfunction with eigenvalue ##1##

Intuition says ##G(x)## should be an eigenfunction, I can't think how to show it from the above however.

Thanks