How to prove that {$sin(x) , sin(2x) , sin(3x) ,…,sin(nx)$} is independent in $mathbb{R}$? [duplicate]












0















This question already has an answer here:




  • How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [on hold]

    5 answers





Prove that {$sin(x) , sin(2x) , sin(3x) ,...,sin(nx)$} is independent in $mathbb{R}$




my trial :



we know that the Wronsekian shouldn't be $0$ to get the trivial solution and thus they are independent. its not trivial to show that $ W not = 0$



W =



begin{vmatrix}
(1)sin(x) & (1)sin(2x) & (1)sin(3x) & ... & (1)sin(nx) \
(1)cos(x) & (2)cos(2x) & (3)cos(3x) & ... & (n)cos(nx) \
-(1)^2sin(x) & -(2)^2sin(2x) & -(3)^2sin(3x) & ... & -(n)^2sin(nx) \
-(1)^3cos(x) & -(2)^3cos(2x) & -(3)^3cos(3x) & ... & -(n)^3cos(nx) \
end{vmatrix}



and so on. it looks like Vandermonde matrix but i cant prove that and so we conclude that its $Wnot =0$






differential-equations







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marked as duplicate by Martin R, TheSimpliFire, 5xum, Shubham Johri, Ahmad Bazzi 16 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • Note that $LaTeX$ commands $sin$ sin and $cos$ cos for future reference.
    – TheSimpliFire
    16 hours ago










  • thanks i updated it
    – Mather
    16 hours ago










  • this is ODE class i don't know whats the topic there but it doesn't like ODE class
    – Mather
    16 hours ago










  • @MartinR And of course that one deserves closure as well
    – TheSimpliFire
    16 hours ago






  • 1




    @Mather The answer I have given does not use any theorem. It is extremely elementary.
    – Kavi Rama Murthy
    16 hours ago
















0















This question already has an answer here:




  • How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [on hold]

    5 answers





Prove that {$sin(x) , sin(2x) , sin(3x) ,...,sin(nx)$} is independent in $mathbb{R}$




my trial :



we know that the Wronsekian shouldn't be $0$ to get the trivial solution and thus they are independent. its not trivial to show that $ W not = 0$



W =



begin{vmatrix}
(1)sin(x) & (1)sin(2x) & (1)sin(3x) & ... & (1)sin(nx) \
(1)cos(x) & (2)cos(2x) & (3)cos(3x) & ... & (n)cos(nx) \
-(1)^2sin(x) & -(2)^2sin(2x) & -(3)^2sin(3x) & ... & -(n)^2sin(nx) \
-(1)^3cos(x) & -(2)^3cos(2x) & -(3)^3cos(3x) & ... & -(n)^3cos(nx) \
end{vmatrix}



and so on. it looks like Vandermonde matrix but i cant prove that and so we conclude that its $Wnot =0$






differential-equations







share|cite|improve this question















marked as duplicate by Martin R, TheSimpliFire, 5xum, Shubham Johri, Ahmad Bazzi 16 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • Note that $LaTeX$ commands $sin$ sin and $cos$ cos for future reference.
    – TheSimpliFire
    16 hours ago










  • thanks i updated it
    – Mather
    16 hours ago










  • this is ODE class i don't know whats the topic there but it doesn't like ODE class
    – Mather
    16 hours ago










  • @MartinR And of course that one deserves closure as well
    – TheSimpliFire
    16 hours ago






  • 1




    @Mather The answer I have given does not use any theorem. It is extremely elementary.
    – Kavi Rama Murthy
    16 hours ago














0












0








0








This question already has an answer here:




  • How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [on hold]

    5 answers





Prove that {$sin(x) , sin(2x) , sin(3x) ,...,sin(nx)$} is independent in $mathbb{R}$




my trial :



we know that the Wronsekian shouldn't be $0$ to get the trivial solution and thus they are independent. its not trivial to show that $ W not = 0$



W =



begin{vmatrix}
(1)sin(x) & (1)sin(2x) & (1)sin(3x) & ... & (1)sin(nx) \
(1)cos(x) & (2)cos(2x) & (3)cos(3x) & ... & (n)cos(nx) \
-(1)^2sin(x) & -(2)^2sin(2x) & -(3)^2sin(3x) & ... & -(n)^2sin(nx) \
-(1)^3cos(x) & -(2)^3cos(2x) & -(3)^3cos(3x) & ... & -(n)^3cos(nx) \
end{vmatrix}



and so on. it looks like Vandermonde matrix but i cant prove that and so we conclude that its $Wnot =0$






differential-equations







share|cite|improve this question
















This question already has an answer here:




  • How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [on hold]

    5 answers





Prove that {$sin(x) , sin(2x) , sin(3x) ,...,sin(nx)$} is independent in $mathbb{R}$




my trial :



we know that the Wronsekian shouldn't be $0$ to get the trivial solution and thus they are independent. its not trivial to show that $ W not = 0$



W =



begin{vmatrix}
(1)sin(x) & (1)sin(2x) & (1)sin(3x) & ... & (1)sin(nx) \
(1)cos(x) & (2)cos(2x) & (3)cos(3x) & ... & (n)cos(nx) \
-(1)^2sin(x) & -(2)^2sin(2x) & -(3)^2sin(3x) & ... & -(n)^2sin(nx) \
-(1)^3cos(x) & -(2)^3cos(2x) & -(3)^3cos(3x) & ... & -(n)^3cos(nx) \
end{vmatrix}



and so on. it looks like Vandermonde matrix but i cant prove that and so we conclude that its $Wnot =0$





This question already has an answer here:




  • How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [on hold]

    5 answers







differential-equations




differential-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 14 hours ago









wilsonw

467315




467315










asked 16 hours ago









Mather

1297




1297




marked as duplicate by Martin R, TheSimpliFire, 5xum, Shubham Johri, Ahmad Bazzi 16 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






marked as duplicate by Martin R, TheSimpliFire, 5xum, Shubham Johri, Ahmad Bazzi 16 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • Note that $LaTeX$ commands $sin$ sin and $cos$ cos for future reference.
    – TheSimpliFire
    16 hours ago










  • thanks i updated it
    – Mather
    16 hours ago










  • this is ODE class i don't know whats the topic there but it doesn't like ODE class
    – Mather
    16 hours ago










  • @MartinR And of course that one deserves closure as well
    – TheSimpliFire
    16 hours ago






  • 1




    @Mather The answer I have given does not use any theorem. It is extremely elementary.
    – Kavi Rama Murthy
    16 hours ago


















  • Note that $LaTeX$ commands $sin$ sin and $cos$ cos for future reference.
    – TheSimpliFire
    16 hours ago










  • thanks i updated it
    – Mather
    16 hours ago










  • this is ODE class i don't know whats the topic there but it doesn't like ODE class
    – Mather
    16 hours ago










  • @MartinR And of course that one deserves closure as well
    – TheSimpliFire
    16 hours ago






  • 1




    @Mather The answer I have given does not use any theorem. It is extremely elementary.
    – Kavi Rama Murthy
    16 hours ago
















Note that $LaTeX$ commands $sin$ sin and $cos$ cos for future reference.
– TheSimpliFire
16 hours ago




Note that $LaTeX$ commands $sin$ sin and $cos$ cos for future reference.
– TheSimpliFire
16 hours ago












thanks i updated it
– Mather
16 hours ago




thanks i updated it
– Mather
16 hours ago












this is ODE class i don't know whats the topic there but it doesn't like ODE class
– Mather
16 hours ago




this is ODE class i don't know whats the topic there but it doesn't like ODE class
– Mather
16 hours ago












@MartinR And of course that one deserves closure as well
– TheSimpliFire
16 hours ago




@MartinR And of course that one deserves closure as well
– TheSimpliFire
16 hours ago




1




1




@Mather The answer I have given does not use any theorem. It is extremely elementary.
– Kavi Rama Murthy
16 hours ago




@Mather The answer I have given does not use any theorem. It is extremely elementary.
– Kavi Rama Murthy
16 hours ago










1 Answer
1






active

oldest

votes


















2














If $sum_{j=1}^{n} c_j sin (jx)$=0 for all $x$ you can prove that each $c_k=0$ by multiplying both sides by $sin(kx)$ and and integrating from $-pi$ to $pi$. Use the fact that $int_{-pi}^{pi} sin(jx)sin (kx), dx =0$ if $j neq k$. This proves the stronger result that the sequence is independent on $[-pi,pi]$.






share|cite|improve this answer























  • This question has been asked and answered before ...
    – Martin R
    16 hours ago










  • thank you but is there an answer related to ODE and wronsekian
    – Mather
    16 hours ago










  • @Mather please emphasize in your question that you would like a Wronsekian oriented answer.
    – Ahmad Bazzi
    16 hours ago










  • the solution is simple as they mentioned above , yet the integral above proves that they are independent only in $ [-pi,pi] $ right ? can we do that for $Re$ in a similar way ?
    – Mather
    15 hours ago








  • 1




    @Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
    – Kavi Rama Murthy
    15 hours ago


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2














If $sum_{j=1}^{n} c_j sin (jx)$=0 for all $x$ you can prove that each $c_k=0$ by multiplying both sides by $sin(kx)$ and and integrating from $-pi$ to $pi$. Use the fact that $int_{-pi}^{pi} sin(jx)sin (kx), dx =0$ if $j neq k$. This proves the stronger result that the sequence is independent on $[-pi,pi]$.






share|cite|improve this answer























  • This question has been asked and answered before ...
    – Martin R
    16 hours ago










  • thank you but is there an answer related to ODE and wronsekian
    – Mather
    16 hours ago










  • @Mather please emphasize in your question that you would like a Wronsekian oriented answer.
    – Ahmad Bazzi
    16 hours ago










  • the solution is simple as they mentioned above , yet the integral above proves that they are independent only in $ [-pi,pi] $ right ? can we do that for $Re$ in a similar way ?
    – Mather
    15 hours ago








  • 1




    @Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
    – Kavi Rama Murthy
    15 hours ago
















2














If $sum_{j=1}^{n} c_j sin (jx)$=0 for all $x$ you can prove that each $c_k=0$ by multiplying both sides by $sin(kx)$ and and integrating from $-pi$ to $pi$. Use the fact that $int_{-pi}^{pi} sin(jx)sin (kx), dx =0$ if $j neq k$. This proves the stronger result that the sequence is independent on $[-pi,pi]$.






share|cite|improve this answer























  • This question has been asked and answered before ...
    – Martin R
    16 hours ago










  • thank you but is there an answer related to ODE and wronsekian
    – Mather
    16 hours ago










  • @Mather please emphasize in your question that you would like a Wronsekian oriented answer.
    – Ahmad Bazzi
    16 hours ago










  • the solution is simple as they mentioned above , yet the integral above proves that they are independent only in $ [-pi,pi] $ right ? can we do that for $Re$ in a similar way ?
    – Mather
    15 hours ago








  • 1




    @Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
    – Kavi Rama Murthy
    15 hours ago














2












2








2






If $sum_{j=1}^{n} c_j sin (jx)$=0 for all $x$ you can prove that each $c_k=0$ by multiplying both sides by $sin(kx)$ and and integrating from $-pi$ to $pi$. Use the fact that $int_{-pi}^{pi} sin(jx)sin (kx), dx =0$ if $j neq k$. This proves the stronger result that the sequence is independent on $[-pi,pi]$.






share|cite|improve this answer














If $sum_{j=1}^{n} c_j sin (jx)$=0 for all $x$ you can prove that each $c_k=0$ by multiplying both sides by $sin(kx)$ and and integrating from $-pi$ to $pi$. Use the fact that $int_{-pi}^{pi} sin(jx)sin (kx), dx =0$ if $j neq k$. This proves the stronger result that the sequence is independent on $[-pi,pi]$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 15 hours ago

























answered 16 hours ago









Kavi Rama Murthy

50.7k31854




50.7k31854












  • This question has been asked and answered before ...
    – Martin R
    16 hours ago










  • thank you but is there an answer related to ODE and wronsekian
    – Mather
    16 hours ago










  • @Mather please emphasize in your question that you would like a Wronsekian oriented answer.
    – Ahmad Bazzi
    16 hours ago










  • the solution is simple as they mentioned above , yet the integral above proves that they are independent only in $ [-pi,pi] $ right ? can we do that for $Re$ in a similar way ?
    – Mather
    15 hours ago








  • 1




    @Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
    – Kavi Rama Murthy
    15 hours ago


















  • This question has been asked and answered before ...
    – Martin R
    16 hours ago










  • thank you but is there an answer related to ODE and wronsekian
    – Mather
    16 hours ago










  • @Mather please emphasize in your question that you would like a Wronsekian oriented answer.
    – Ahmad Bazzi
    16 hours ago










  • the solution is simple as they mentioned above , yet the integral above proves that they are independent only in $ [-pi,pi] $ right ? can we do that for $Re$ in a similar way ?
    – Mather
    15 hours ago








  • 1




    @Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
    – Kavi Rama Murthy
    15 hours ago
















This question has been asked and answered before ...
– Martin R
16 hours ago




This question has been asked and answered before ...
– Martin R
16 hours ago












thank you but is there an answer related to ODE and wronsekian
– Mather
16 hours ago




thank you but is there an answer related to ODE and wronsekian
– Mather
16 hours ago












@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
– Ahmad Bazzi
16 hours ago




@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
– Ahmad Bazzi
16 hours ago












the solution is simple as they mentioned above , yet the integral above proves that they are independent only in $ [-pi,pi] $ right ? can we do that for $Re$ in a similar way ?
– Mather
15 hours ago






the solution is simple as they mentioned above , yet the integral above proves that they are independent only in $ [-pi,pi] $ right ? can we do that for $Re$ in a similar way ?
– Mather
15 hours ago






1




1




@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
– Kavi Rama Murthy
15 hours ago




@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
– Kavi Rama Murthy
15 hours ago



-

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