Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $f(x)=(\sin x)^2$ and $g=(\cos x)^2$, would $\cos(2x)$ lie in space spanned by $f$ and $g$. What about $3+x$?

share|improve this question
1  
What have you tried? –  Zach L. Dec 13 '12 at 16:24
2  
Take a look at the double angle formulas. Convince yourself that a linear combination of two bounded functions is bounded, and of course $\sin^2,\cos^2$ are both bounded - but $3+x$ is not bounded. Finally, this has nothing to do with Mathematica. –  anon Dec 13 '12 at 16:24
    
@ZachL. I know two conditions have to satisfy, 1-> If U and V are in vector space V, then U+V should also be in that V. 2-> If k is non-negative scalar then KU should also lie in vector space. I just don't know how to implement this concept on functions but I do know how it works on vectors –  Terrenium Dec 13 '12 at 16:30
    
@anon sorry, too early for me (last comment deleted) –  David Mitra Dec 13 '12 at 16:30
    
I just did give a mathematical explanation of how to answer your questions - or perhaps I misunderstand what you mean by "it" in your latest comment. (Also, why is this being downvoted? It isn't a great question, but I do not feel OP is being obstinate or anything deserving a -1.) –  anon Dec 13 '12 at 16:37

1 Answer 1

Can you write $\cos 2x$ as a linear combination of $\cos^2 x$ and $\sin^2 x$? So what can you say about whether or not it lies in the span?

As for $3+x$, consider instead the following:

If $f$ and $g$ are periodic (with the same period, $P$ say) then, for all scalars $a,b$, $af+bg$ is also periodic with period $P$.

Prove this first, and then deduce whether or not $3+x$ lies in the span.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.