Legendre's proof involving linearity independence
0
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Show that any polynomial of degree n is a linear combination of P0(x), P1(x), ..., Pn(x)
Actually I have no idea how to start with a proof involving "any". Can someone help??
legendre-symbol
asked Jan 18 at 13:34
user635977user635977
63
$endgroup$
add a comment |
0
$begingroup$
Show that any polynomial of degree n is a linear combination of P0(x), P1(x), ..., Pn(x)
Actually I have no idea how to start with a proof involving "any". Can someone help??
legendre-symbol
asked Jan 18 at 13:34
user635977user635977
63
$endgroup$
$begingroup$
See one of my other posts on this subject: math.stackexchange.com/questions/512295/…
$endgroup$
– Maestro13
Feb 14 at 23:06
add a comment |
0
0
0
$begingroup$
Show that any polynomial of degree n is a linear combination of P0(x), P1(x), ..., Pn(x)
Actually I have no idea how to start with a proof involving "any". Can someone help??
legendre-symbol
asked Jan 18 at 13:34
user635977user635977
63
$endgroup$
Show that any polynomial of degree n is a linear combination of P0(x), P1(x), ..., Pn(x)
Actually I have no idea how to start with a proof involving "any". Can someone help??
legendre-symbol
legendre-symbol
asked Jan 18 at 13:34
user635977user635977
63
asked Jan 18 at 13:34
user635977user635977
63
asked Jan 18 at 13:34
user635977user635977
63
asked Jan 18 at 13:34
user635977user635977
63
asked Jan 18 at 13:34
user635977user635977
63
63
$begingroup$
See one of my other posts on this subject: math.stackexchange.com/questions/512295/…
$endgroup$
– Maestro13
Feb 14 at 23:06
add a comment |
$begingroup$
See one of my other posts on this subject: math.stackexchange.com/questions/512295/…
$endgroup$
– Maestro13
Feb 14 at 23:06
$begingroup$
See one of my other posts on this subject: math.stackexchange.com/questions/512295/…
$endgroup$
– Maestro13
Feb 14 at 23:06
$begingroup$
See one of my other posts on this subject: math.stackexchange.com/questions/512295/…
$endgroup$
– Maestro13
Feb 14 at 23:06
add a comment |
1 Answer
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gram schmidt orthogonalization process under inner product$$ int _{-1}^{1} f(x)g(x)dx$$ may help.
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
$endgroup$
$begingroup$
another approach may be from legendre equation
$endgroup$
– Bijayan Ray
Jan 18 at 13:53
$begingroup$
can you elaborate more?
$endgroup$
– user635977
Jan 19 at 17:20
$begingroup$
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent
$endgroup$
– Bijayan Ray
Jan 20 at 12:19
add a comment |
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1 Answer
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1 Answer
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active
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0
$begingroup$
gram schmidt orthogonalization process under inner product$$ int _{-1}^{1} f(x)g(x)dx$$ may help.
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
$endgroup$
$begingroup$
another approach may be from legendre equation
$endgroup$
– Bijayan Ray
Jan 18 at 13:53
$begingroup$
can you elaborate more?
$endgroup$
– user635977
Jan 19 at 17:20
$begingroup$
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent
$endgroup$
– Bijayan Ray
Jan 20 at 12:19
add a comment |
0
$begingroup$
gram schmidt orthogonalization process under inner product$$ int _{-1}^{1} f(x)g(x)dx$$ may help.
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
$endgroup$
$begingroup$
another approach may be from legendre equation
$endgroup$
– Bijayan Ray
Jan 18 at 13:53
$begingroup$
can you elaborate more?
$endgroup$
– user635977
Jan 19 at 17:20
$begingroup$
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent
$endgroup$
– Bijayan Ray
Jan 20 at 12:19
add a comment |
0
0
0
$begingroup$
gram schmidt orthogonalization process under inner product$$ int _{-1}^{1} f(x)g(x)dx$$ may help.
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
$endgroup$
gram schmidt orthogonalization process under inner product$$ int _{-1}^{1} f(x)g(x)dx$$ may help.
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
answered Jan 18 at 13:52
Bijayan RayBijayan Ray
1511213
1511213
$begingroup$
another approach may be from legendre equation
$endgroup$
– Bijayan Ray
Jan 18 at 13:53
$begingroup$
can you elaborate more?
$endgroup$
– user635977
Jan 19 at 17:20
$begingroup$
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent
$endgroup$
– Bijayan Ray
Jan 20 at 12:19
add a comment |
$begingroup$
another approach may be from legendre equation
$endgroup$
– Bijayan Ray
Jan 18 at 13:53
$begingroup$
can you elaborate more?
$endgroup$
– user635977
Jan 19 at 17:20
$begingroup$
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent
$endgroup$
– Bijayan Ray
Jan 20 at 12:19
$begingroup$
another approach may be from legendre equation
$endgroup$
– Bijayan Ray
Jan 18 at 13:53
$begingroup$
another approach may be from legendre equation
$endgroup$
– Bijayan Ray
Jan 18 at 13:53
$begingroup$
can you elaborate more?
$endgroup$
– user635977
Jan 19 at 17:20
$begingroup$
can you elaborate more?
$endgroup$
– user635977
Jan 19 at 17:20
$begingroup$
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent
$endgroup$
– Bijayan Ray
Jan 20 at 12:19
$begingroup$
basically on applying gram schmidt orthogonalization process under given inner product on P(x) would give some scalar multiple of P0(x), P1(x), ..., Pn(x) implying that they are orthogonal hence linearly independent
$endgroup$
– Bijayan Ray
Jan 20 at 12:19
add a comment |
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$begingroup$
See one of my other posts on this subject: math.stackexchange.com/questions/512295/…
$endgroup$
– Maestro13
Feb 14 at 23:06