How to prove that {$sin(x) , sin(2x) , sin(3x) ,…,sin(nx)$} is independent in $mathbb{R}$? [duplicate]
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This question already has an answer here:
How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [closed]
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 $Wnot =0$
ordinary-differential-equations
edited Jan 3 at 11:00
wilsonw
478315
asked Jan 3 at 9:07
Mather Mather
4008
$endgroup$
marked as duplicate by Martin R, TheSimpliFire, 5xum, Shubham Johri, Ahmad Bazzi Jan 3 at 9:17
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.
|
show 5 more comments
$begingroup$
This question already has an answer here:
How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [closed]
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 $Wnot =0$
ordinary-differential-equations
edited Jan 3 at 11:00
wilsonw
478315
asked Jan 3 at 9:07
Mather Mather
4008
$endgroup$
marked as duplicate by Martin R, TheSimpliFire, 5xum, Shubham Johri, Ahmad Bazzi Jan 3 at 9:17
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.
$begingroup$
Note that $LaTeX$ commands $sin$sinand $cos$cosfor future reference.
$endgroup$
– TheSimpliFire
Jan 3 at 9:09
$begingroup$
thanks i updated it
$endgroup$
– Mather
Jan 3 at 9:10
$begingroup$
this is ODE class i don't know whats the topic there but it doesn't like ODE class
$endgroup$
– Mather
Jan 3 at 9:13
$begingroup$
@MartinR And of course that one deserves closure as well
$endgroup$
– TheSimpliFire
Jan 3 at 9:13
1
$begingroup$
@Mather The answer I have given does not use any theorem. It is extremely elementary.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 9:18
|
show 5 more comments
$begingroup$
This question already has an answer here:
How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [closed]
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 $Wnot =0$
ordinary-differential-equations
edited Jan 3 at 11:00
wilsonw
478315
asked Jan 3 at 9:07
Mather Mather
4008
$endgroup$
This question already has an answer here:
How to prove that the set ${sin(x),sin(2x),…,sin(mx)}$ is linearly independent? [closed]
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? [closed]
5 answers
ordinary-differential-equations
ordinary-differential-equations
edited Jan 3 at 11:00
wilsonw
478315
asked Jan 3 at 9:07
Mather Mather
4008
edited Jan 3 at 11:00
wilsonw
478315
asked Jan 3 at 9:07
Mather Mather
4008
edited Jan 3 at 11:00
wilsonw
478315
edited Jan 3 at 11:00
wilsonw
478315
edited Jan 3 at 11:00
wilsonw
478315
478315
asked Jan 3 at 9:07
Mather Mather
4008
asked Jan 3 at 9:07
Mather Mather
4008
asked Jan 3 at 9:07
Mather Mather
4008
4008
marked as duplicate by Martin R, TheSimpliFire, 5xum, Shubham Johri, Ahmad Bazzi Jan 3 at 9:17
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 Jan 3 at 9:17
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.
$begingroup$
Note that $LaTeX$ commands $sin$sinand $cos$cosfor future reference.
$endgroup$
– TheSimpliFire
Jan 3 at 9:09
$begingroup$
thanks i updated it
$endgroup$
– Mather
Jan 3 at 9:10
$begingroup$
this is ODE class i don't know whats the topic there but it doesn't like ODE class
$endgroup$
– Mather
Jan 3 at 9:13
$begingroup$
@MartinR And of course that one deserves closure as well
$endgroup$
– TheSimpliFire
Jan 3 at 9:13
1
$begingroup$
@Mather The answer I have given does not use any theorem. It is extremely elementary.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 9:18
|
show 5 more comments
$begingroup$
Note that $LaTeX$ commands $sin$sinand $cos$cosfor future reference.
$endgroup$
– TheSimpliFire
Jan 3 at 9:09
$begingroup$
thanks i updated it
$endgroup$
– Mather
Jan 3 at 9:10
$begingroup$
this is ODE class i don't know whats the topic there but it doesn't like ODE class
$endgroup$
– Mather
Jan 3 at 9:13
$begingroup$
@MartinR And of course that one deserves closure as well
$endgroup$
– TheSimpliFire
Jan 3 at 9:13
1
$begingroup$
@Mather The answer I have given does not use any theorem. It is extremely elementary.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 9:18
$begingroup$
Note that $LaTeX$ commands $sin$
sin and $cos$ cos for future reference.$endgroup$
– TheSimpliFire
Jan 3 at 9:09
$begingroup$
Note that $LaTeX$ commands $sin$
sin and $cos$ cos for future reference.$endgroup$
– TheSimpliFire
Jan 3 at 9:09
$begingroup$
thanks i updated it
$endgroup$
– Mather
Jan 3 at 9:10
$begingroup$
thanks i updated it
$endgroup$
– Mather
Jan 3 at 9:10
$begingroup$
this is ODE class i don't know whats the topic there but it doesn't like ODE class
$endgroup$
– Mather
Jan 3 at 9:13
$begingroup$
this is ODE class i don't know whats the topic there but it doesn't like ODE class
$endgroup$
– Mather
Jan 3 at 9:13
$begingroup$
@MartinR And of course that one deserves closure as well
$endgroup$
– TheSimpliFire
Jan 3 at 9:13
$begingroup$
@MartinR And of course that one deserves closure as well
$endgroup$
– TheSimpliFire
Jan 3 at 9:13
1
1
$begingroup$
@Mather The answer I have given does not use any theorem. It is extremely elementary.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 9:18
$begingroup$
@Mather The answer I have given does not use any theorem. It is extremely elementary.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 9:18
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
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 $[-pi,pi]$.
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
$endgroup$
$begingroup$
This question has been asked and answered before ...
$endgroup$
– Martin R
Jan 3 at 9:13
$begingroup$
thank you but is there an answer related to ODE and wronsekian
$endgroup$
– Mather
Jan 3 at 9:16
$begingroup$
@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
$endgroup$
– Ahmad Bazzi
Jan 3 at 9:18
$begingroup$
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 ?
$endgroup$
– Mather
Jan 3 at 9:58
1
$begingroup$
@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 10:00
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 $[-pi,pi]$.
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
$endgroup$
$begingroup$
This question has been asked and answered before ...
$endgroup$
– Martin R
Jan 3 at 9:13
$begingroup$
thank you but is there an answer related to ODE and wronsekian
$endgroup$
– Mather
Jan 3 at 9:16
$begingroup$
@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
$endgroup$
– Ahmad Bazzi
Jan 3 at 9:18
$begingroup$
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 ?
$endgroup$
– Mather
Jan 3 at 9:58
1
$begingroup$
@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 10:00
add a comment |
$begingroup$
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 $[-pi,pi]$.
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
$endgroup$
$begingroup$
This question has been asked and answered before ...
$endgroup$
– Martin R
Jan 3 at 9:13
$begingroup$
thank you but is there an answer related to ODE and wronsekian
$endgroup$
– Mather
Jan 3 at 9:16
$begingroup$
@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
$endgroup$
– Ahmad Bazzi
Jan 3 at 9:18
$begingroup$
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 ?
$endgroup$
– Mather
Jan 3 at 9:58
1
$begingroup$
@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 10:00
add a comment |
$begingroup$
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 $[-pi,pi]$.
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
$endgroup$
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 $[-pi,pi]$.
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
edited Jan 3 at 10:02
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
answered Jan 3 at 9:12
Kavi Rama MurthyKavi Rama Murthy
65.8k42867
65.8k42867
$begingroup$
This question has been asked and answered before ...
$endgroup$
– Martin R
Jan 3 at 9:13
$begingroup$
thank you but is there an answer related to ODE and wronsekian
$endgroup$
– Mather
Jan 3 at 9:16
$begingroup$
@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
$endgroup$
– Ahmad Bazzi
Jan 3 at 9:18
$begingroup$
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 ?
$endgroup$
– Mather
Jan 3 at 9:58
1
$begingroup$
@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 10:00
add a comment |
$begingroup$
This question has been asked and answered before ...
$endgroup$
– Martin R
Jan 3 at 9:13
$begingroup$
thank you but is there an answer related to ODE and wronsekian
$endgroup$
– Mather
Jan 3 at 9:16
$begingroup$
@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
$endgroup$
– Ahmad Bazzi
Jan 3 at 9:18
$begingroup$
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 ?
$endgroup$
– Mather
Jan 3 at 9:58
1
$begingroup$
@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 10:00
$begingroup$
This question has been asked and answered before ...
$endgroup$
– Martin R
Jan 3 at 9:13
$begingroup$
This question has been asked and answered before ...
$endgroup$
– Martin R
Jan 3 at 9:13
$begingroup$
thank you but is there an answer related to ODE and wronsekian
$endgroup$
– Mather
Jan 3 at 9:16
$begingroup$
thank you but is there an answer related to ODE and wronsekian
$endgroup$
– Mather
Jan 3 at 9:16
$begingroup$
@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
$endgroup$
– Ahmad Bazzi
Jan 3 at 9:18
$begingroup$
@Mather please emphasize in your question that you would like a Wronsekian oriented answer.
$endgroup$
– Ahmad Bazzi
Jan 3 at 9:18
$begingroup$
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 ?
$endgroup$
– Mather
Jan 3 at 9:58
$begingroup$
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 ?
$endgroup$
– Mather
Jan 3 at 9:58
1
1
$begingroup$
@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 10:00
$begingroup$
@Mather Independent on a subset of $mathbb R$ implies indepedence on all of $mathbb R$.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 10:00
add a comment |
$begingroup$
Note that $LaTeX$ commands $sin$
sinand $cos$cosfor future reference.$endgroup$
– TheSimpliFire
Jan 3 at 9:09
$begingroup$
thanks i updated it
$endgroup$
– Mather
Jan 3 at 9:10
$begingroup$
this is ODE class i don't know whats the topic there but it doesn't like ODE class
$endgroup$
– Mather
Jan 3 at 9:13
$begingroup$
@MartinR And of course that one deserves closure as well
$endgroup$
– TheSimpliFire
Jan 3 at 9:13
1
$begingroup$
@Mather The answer I have given does not use any theorem. It is extremely elementary.
$endgroup$
– Kavi Rama Murthy
Jan 3 at 9:18