Linear Independence of trigonometric functions

  • #1

Homework Statement


There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x.

Homework Equations


s1sin2xcos2x+s2sin2x+s3cos2x+s4sin2x+s5cos2x=0
I need to prove that all s1,s2,s3,s4 and s5 must be equal to zero for the above equation to be true.

The Attempt at a Solution


I used the trigonometric formulas and came to this:
(s1/2 + s2)sin2x + (s3-s4/2 + s5/2)cos2x + [(s4+s5)/2] = 0.
We usually use the derive here but it doesn't seem to help.

Edit: Oh, yeah. We haven't been taught the matrix yet.
 
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Answers and Replies

  • #2
S.G. Janssens
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There are probably easier ways, but you could show that the associated Wronskian doesn't vanish identically on ##\mathbb{R}##.

EDIT: Ok, I see you edited your message. In that case the Wronskian is probably not the way to go.
 
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  • #3
S.G. Janssens
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Hold on... They are not linearly independent, because ##\cos{2x} = \cos^2{x} - \sin^2{x}##.
 
  • #4
Well... I made that assumption because I can't think of another way to solve the problem.
The problem asks to find the dimension of the subspace of space C(-π,π) that is produced by these functions.
I thought that if these vectors-functions are linearly independent then they are a base of the subspace, and that's how I prove its dimension.
 
  • #5
S.G. Janssens
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The hint I can give you is to set up the condition
s1sin2xcos2x+s2sin2x+s3cos2x+s4sin2x+s5cos2x=0
but leave the ##\cos{2x}## term out, because we already know that this is a linear combination of some of the other functions. The dimension of the subspace of interest is therefore at most..?

Then evaluate this condition in some well-chosen points in ##(-\pi,\pi)## such as: ##x = 0, x = \frac{\pi}{2}## and some others. This will give you sufficiently many independent linear equations from which you can solve for the ##s_i## to draw your conclusion.
 
  • #6
It's late and I've been working on the problem for a long time now. I'll review your hint tomorrow and let you know. Thanks a lot!
Although it doesn't look like the way we work, I'm going to use it as long as I understand it.
 
  • #7
S.G. Janssens
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Ok, then we both go to sleep. Let me know how it worked, good luck!
 
  • #8
If I prove that at least one of them is a linear combination of the others, does this mean that the same subspace can be produced without this one vector?
 
  • #9
S.G. Janssens
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If I prove that at least one of them is a linear combination of the others, does this mean that the same subspace can be produced without this one vector?
Yes. Maybe after you have used this fact to solve your exercise, you could try to prove that as well.

For that, you would assume that you have vectors ##\{v_1,\ldots,v_m\}## in a vector space and, furthermore, there is some ##1 \le k \le m## such that ##v_k## is a linear combination of the other vectors. Then it would be up to you to show that any linear combination of ##v_1,\ldots,v_m## can be written as a linear combination of ##v_1,\ldots,v_m## with ##v_k## excluded.

If you find the different indices confusing, first try it with, say, three vectors, the last of them being a linear combination of the first two.
 
  • #10
I don't really mind the proofs right now - they are all written in my notebook. The course itself doesn't even mind the proofs.
So, if I keep doing this (excluding a vector that is a linear combination of another) until I end up with n vectors that are linearly independent, I have a base of the subspace. And the number n is its dimension. Am I right?
 
  • #11
S.G. Janssens
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Am I right?
Yes. Still, if you have time, I would recommend that you try to prove these things for yourself, from scratch. If the proofs are already in your (note)book, even better, then you can check yourself. When you practise these kinds of small proofs, it also becomes easier to do the regular exercises.
 
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  • #12
Thanks for your answers and advise. I'll find some time on the weekend to prove it.
 
  • #13
Hmm... is sin2x a linear combination of cos2x? I know that cos2x=sin(2x+π/2) but I'm not sure if that's a linear combination. Same goes for √(1-sin²2x).
 
  • #14
S.G. Janssens
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Carefully review the definition of "linear combination", this is important.

I can already tell you that ##\sin{2x}## is not a linear combination (i.e. a multiple, because we only have one vector) of ##\cos{2x}## on ##(-\pi,\pi)##. Why not? Loosely speaking, because the graph of ##\sin{2x}## is not a scalar multiple of the graph of ##\cos{2x}##.Indeed, ##\sin{\frac{\pi}{2}} = 1## and ##\cos{\frac{\pi}{2}} = 0## so there can never be a scalar ##c## such that
$$
\sin{2x} = c\cos{2x} \qquad \forall\,x \in (-\pi,\pi) \qquad \text{(not true!)}
$$
For the other one, note that ##\sqrt{1 - \sin^2{2x}} = |\cos{2x}|## and this is not a linear combination of ##\cos{2x}## on ##(-\pi,\pi)##, nor of ##\sin{2x}##, nor the other way around.
 
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  • #15
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Hmm... is sin2x a linear combination of cos2x?

Carefully review the definition of "linear combination", this is important.
I agree.
It's very easy (in fact, almost trivial) to determine whether a vector v is a linear combination of another vector u: v will be a scalar multiple of u. It's not so easy to determine whether one vector is a linear combination of two or more other vectors.
 
  • #16
But I can't find any way to prove it using the methods we've been taught. I used again the above condition and its derivative with all possible combinations and none worked.
 
  • #17
HallsofIvy
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Then what definition of "linear combination" have you been taught? That, together with trig identities, should be enough.
 
  • #18
S.G. Janssens
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I gave you this hint a few days ago and I still think it may be the easiest way. I edited the quote a bit to make it more explicit (and also to fix the weird way formulas are sometimes copied into quotes).

The hint I can give you is to set up the condition
$$
s_1\sin{2x}\cos{2x}+s_2\sin{2x}+s_4\sin^2{x}+s_5\cos^2{x}=0 \qquad (*)
$$
llike you already did, but now with the term ##s_3 \cos{2x}## left out, because we already know that ##\cos{2x}## is a linear combination of some of the other functions. The dimension of the subspace of interest is therefore at most 4.

Then evaluate (*) in some well-chosen points in ##(-\pi,\pi)## such as: ##x = 0, x = \frac{\pi}{2}## and two others. This will give you sufficiently many independent linear equations from which you can solve for ##s_1 = s_2 = s_4 = s_5 = 0## to draw your conclusion.

It is however important that you do not just carry this out, but also understand precisely what this has to do with linear independence and why it solves your problem.
 
  • #19
Okay, first of all, I made a mistake when I first copied the problem. The first function is sinxcosx, not sin2xcos2x.

Now, let me take it step by step, because I've gone further than the condition you suggest I should prove, and that's probably a mistake. I also left out sinxcosx(=sin2x/2), as well as cos²x(=1-sin²x), exactly the way I described earlier
if I keep doing this (excluding a vector that is a linear combination of another) until I end up with n vectors that are linearly independent, I have a base of the subspace
and ended up with sin2x and cos2x. Now my goal was to prove that these two functions are linearly independent, that's where my message was referring to.

Is this wrong anywhere?
 
  • #20
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Homework Statement


There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x.
##\cos 2x = \cos ^2x - \sin ^2 x##. System is linearly dependent.
 
  • #21
S.G. Janssens
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cos2x=cos2x−sin2x\cos 2x = \cos ^2x - \sin ^2 x. System is linearly dependent.
See Monday.
 
  • #22
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Oops, I just read the problem and it was the first thing that stuck out to me, forgot to look at the replies :(
 
  • #23
S.G. Janssens
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Oops, I just read the problem and it was the first thing that stuck out to me, forgot to look at the replies :(
No problem :wink: Actually, I overlooked it at first, as you can see when you read back.
 
  • #24
S.G. Janssens
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Okay, first of all, I made a mistake when I first copied the problem. The first function is sinxcosx, not sin2xcos2x.
So, just to be sure, can you confirm that in the correct version of the problem we have the functions
$$
\sin{x}\cos{x}, \sin{2x}, \cos{2x}, \sin^2{x}, \cos^2{x}
$$
I also left out ##\sin{x}\cos{x} ( = \sin{2x/2})##, as well as ##\cos^2{x} (=1-\sin^2{x})##, exactly the way I described earlier
I agree to the first elimination (however, I would write ##\frac{1}{2}\sin{2x}## for clarity), but not to the second one. Yes it holds that ##\cos^2{x} = 1-\sin^2{x}## but the constant function "1" is not in your initial set, so from this you cannot conclude dependence of ##\cos^2{x}##.
Is this wrong anywhere?
So far, we have seen we can justly throw out ##\cos{2x}## and ##\sin{2x}##. (The latter could not be thrown out when we were still using the wrong version of your problem statement.) So now you are left with
$$
\sin{x}\cos{x}, \sin^2{x}, \cos^2{x}
$$
and it is up to you to verify whether this smaller set (which spans the same space as your original set), is linearly independent, or whether you can throw away more vectors.
 
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  • #25
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If the objective is the same, then the system is still linearly dependent. For any sub-system's linear dependence implies the entire system is linearly dependent.
 

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