With $lambda_1alpha_1+lambda_2alpha_2+cdots+lambda_nalpha_m=0$,how to show the existence of such an $alpha$...
$begingroup$
Let V be an n-dimensional inner product space over the field $mathbb{R}$.Suppose vectors $alpha_1,alpha_2,cdots,alpha_min V$ satisfy the following property:If there exist nonegative real numbers $lambda_1,lambda_2,cdots,lambda_m$such that $lambda_1alpha_1+lambda_2alpha_2+cdots+lambda_nalpha_m=0$,then $lambda_1=lambda_2=cdots=lambda_m=0$.
Try to prove that there exists such an $alpha in V$ that $(alpha,alpha_i)>0$ for all $i=1,cdots,m$.
I have tried hard but without any progress.If the set $alpha_1,alpha_2,cdots,alpha_m$ is linear independent,we can use the Schmidt's progress to show the existence of $alpha$.Indeed,we let
begin{align*}
beta_1=&alpha_1,\
beta_k=&alpha_k-sum_{j<k}frac{(alpha_k,beta_j)}{(beta_j,beta_j)}beta_jtriangleq alpha_k-sum_{j<k}C_{jk}beta_j,~~k=2,cdots,m
end{align*}
then the vectors $beta_1,beta_2,cdots,beta_m$are orthogonal to each other.Let
begin{align*}
gamma_1=&alpha_1\
gamma_k=&gamma_{k-1}+mu_k beta_k~~k=2,cdots,m
end{align*}
where all ${mu_k}'s$ will be determined.Let $alpha=gamma_m$,then
begin{align*}
(alpha,alpha_k)=(gamma_m,alpha_k)=(gamma_m,beta_k+sum_{j<k}C_{jk}beta_j)
=mu_k+sum_{j<k}C_{jk}mu_j
end{align*}
So we can determine all ${mu_k}'s$ Recursively:
begin{align*}
mu_1=1,mu_k>-sum_{j<k}C_{jk}mu_j,
end{align*}
and such an $alpha$ is what we need.
linear-algebra
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
$endgroup$
|
show 1 more comment
$begingroup$
Let V be an n-dimensional inner product space over the field $mathbb{R}$.Suppose vectors $alpha_1,alpha_2,cdots,alpha_min V$ satisfy the following property:If there exist nonegative real numbers $lambda_1,lambda_2,cdots,lambda_m$such that $lambda_1alpha_1+lambda_2alpha_2+cdots+lambda_nalpha_m=0$,then $lambda_1=lambda_2=cdots=lambda_m=0$.
Try to prove that there exists such an $alpha in V$ that $(alpha,alpha_i)>0$ for all $i=1,cdots,m$.
I have tried hard but without any progress.If the set $alpha_1,alpha_2,cdots,alpha_m$ is linear independent,we can use the Schmidt's progress to show the existence of $alpha$.Indeed,we let
begin{align*}
beta_1=&alpha_1,\
beta_k=&alpha_k-sum_{j<k}frac{(alpha_k,beta_j)}{(beta_j,beta_j)}beta_jtriangleq alpha_k-sum_{j<k}C_{jk}beta_j,~~k=2,cdots,m
end{align*}
then the vectors $beta_1,beta_2,cdots,beta_m$are orthogonal to each other.Let
begin{align*}
gamma_1=&alpha_1\
gamma_k=&gamma_{k-1}+mu_k beta_k~~k=2,cdots,m
end{align*}
where all ${mu_k}'s$ will be determined.Let $alpha=gamma_m$,then
begin{align*}
(alpha,alpha_k)=(gamma_m,alpha_k)=(gamma_m,beta_k+sum_{j<k}C_{jk}beta_j)
=mu_k+sum_{j<k}C_{jk}mu_j
end{align*}
So we can determine all ${mu_k}'s$ Recursively:
begin{align*}
mu_1=1,mu_k>-sum_{j<k}C_{jk}mu_j,
end{align*}
and such an $alpha$ is what we need.
linear-algebra
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
$endgroup$
$begingroup$
What is the relationship between m and n?
$endgroup$
– herb steinberg
Dec 2 '18 at 1:41
$begingroup$
They are irrelevant.
$endgroup$
– mbfkk
Dec 2 '18 at 1:46
$begingroup$
Your statement seems to have a typo. The first and third occurrences have $lambda_m$ while the second has $lambda_n$.
$endgroup$
– herb steinberg
Dec 2 '18 at 1:52
$begingroup$
Yes,I'm sorry .
$endgroup$
– mbfkk
Dec 2 '18 at 1:59
$begingroup$
Please correct!
$endgroup$
– herb steinberg
Dec 2 '18 at 3:43
|
show 1 more comment
$begingroup$
Let V be an n-dimensional inner product space over the field $mathbb{R}$.Suppose vectors $alpha_1,alpha_2,cdots,alpha_min V$ satisfy the following property:If there exist nonegative real numbers $lambda_1,lambda_2,cdots,lambda_m$such that $lambda_1alpha_1+lambda_2alpha_2+cdots+lambda_nalpha_m=0$,then $lambda_1=lambda_2=cdots=lambda_m=0$.
Try to prove that there exists such an $alpha in V$ that $(alpha,alpha_i)>0$ for all $i=1,cdots,m$.
I have tried hard but without any progress.If the set $alpha_1,alpha_2,cdots,alpha_m$ is linear independent,we can use the Schmidt's progress to show the existence of $alpha$.Indeed,we let
begin{align*}
beta_1=&alpha_1,\
beta_k=&alpha_k-sum_{j<k}frac{(alpha_k,beta_j)}{(beta_j,beta_j)}beta_jtriangleq alpha_k-sum_{j<k}C_{jk}beta_j,~~k=2,cdots,m
end{align*}
then the vectors $beta_1,beta_2,cdots,beta_m$are orthogonal to each other.Let
begin{align*}
gamma_1=&alpha_1\
gamma_k=&gamma_{k-1}+mu_k beta_k~~k=2,cdots,m
end{align*}
where all ${mu_k}'s$ will be determined.Let $alpha=gamma_m$,then
begin{align*}
(alpha,alpha_k)=(gamma_m,alpha_k)=(gamma_m,beta_k+sum_{j<k}C_{jk}beta_j)
=mu_k+sum_{j<k}C_{jk}mu_j
end{align*}
So we can determine all ${mu_k}'s$ Recursively:
begin{align*}
mu_1=1,mu_k>-sum_{j<k}C_{jk}mu_j,
end{align*}
and such an $alpha$ is what we need.
linear-algebra
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
$endgroup$
Let V be an n-dimensional inner product space over the field $mathbb{R}$.Suppose vectors $alpha_1,alpha_2,cdots,alpha_min V$ satisfy the following property:If there exist nonegative real numbers $lambda_1,lambda_2,cdots,lambda_m$such that $lambda_1alpha_1+lambda_2alpha_2+cdots+lambda_nalpha_m=0$,then $lambda_1=lambda_2=cdots=lambda_m=0$.
Try to prove that there exists such an $alpha in V$ that $(alpha,alpha_i)>0$ for all $i=1,cdots,m$.
I have tried hard but without any progress.If the set $alpha_1,alpha_2,cdots,alpha_m$ is linear independent,we can use the Schmidt's progress to show the existence of $alpha$.Indeed,we let
begin{align*}
beta_1=&alpha_1,\
beta_k=&alpha_k-sum_{j<k}frac{(alpha_k,beta_j)}{(beta_j,beta_j)}beta_jtriangleq alpha_k-sum_{j<k}C_{jk}beta_j,~~k=2,cdots,m
end{align*}
then the vectors $beta_1,beta_2,cdots,beta_m$are orthogonal to each other.Let
begin{align*}
gamma_1=&alpha_1\
gamma_k=&gamma_{k-1}+mu_k beta_k~~k=2,cdots,m
end{align*}
where all ${mu_k}'s$ will be determined.Let $alpha=gamma_m$,then
begin{align*}
(alpha,alpha_k)=(gamma_m,alpha_k)=(gamma_m,beta_k+sum_{j<k}C_{jk}beta_j)
=mu_k+sum_{j<k}C_{jk}mu_j
end{align*}
So we can determine all ${mu_k}'s$ Recursively:
begin{align*}
mu_1=1,mu_k>-sum_{j<k}C_{jk}mu_j,
end{align*}
and such an $alpha$ is what we need.
linear-algebra
linear-algebra
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
edited Dec 2 '18 at 2:00
mbfkk
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
asked Dec 2 '18 at 1:13
mbfkkmbfkk
336113
336113
$begingroup$
What is the relationship between m and n?
$endgroup$
– herb steinberg
Dec 2 '18 at 1:41
$begingroup$
They are irrelevant.
$endgroup$
– mbfkk
Dec 2 '18 at 1:46
$begingroup$
Your statement seems to have a typo. The first and third occurrences have $lambda_m$ while the second has $lambda_n$.
$endgroup$
– herb steinberg
Dec 2 '18 at 1:52
$begingroup$
Yes,I'm sorry .
$endgroup$
– mbfkk
Dec 2 '18 at 1:59
$begingroup$
Please correct!
$endgroup$
– herb steinberg
Dec 2 '18 at 3:43
|
show 1 more comment
$begingroup$
What is the relationship between m and n?
$endgroup$
– herb steinberg
Dec 2 '18 at 1:41
$begingroup$
They are irrelevant.
$endgroup$
– mbfkk
Dec 2 '18 at 1:46
$begingroup$
Your statement seems to have a typo. The first and third occurrences have $lambda_m$ while the second has $lambda_n$.
$endgroup$
– herb steinberg
Dec 2 '18 at 1:52
$begingroup$
Yes,I'm sorry .
$endgroup$
– mbfkk
Dec 2 '18 at 1:59
$begingroup$
Please correct!
$endgroup$
– herb steinberg
Dec 2 '18 at 3:43
$begingroup$
What is the relationship between m and n?
$endgroup$
– herb steinberg
Dec 2 '18 at 1:41
$begingroup$
What is the relationship between m and n?
$endgroup$
– herb steinberg
Dec 2 '18 at 1:41
$begingroup$
They are irrelevant.
$endgroup$
– mbfkk
Dec 2 '18 at 1:46
$begingroup$
They are irrelevant.
$endgroup$
– mbfkk
Dec 2 '18 at 1:46
$begingroup$
Your statement seems to have a typo. The first and third occurrences have $lambda_m$ while the second has $lambda_n$.
$endgroup$
– herb steinberg
Dec 2 '18 at 1:52
$begingroup$
Your statement seems to have a typo. The first and third occurrences have $lambda_m$ while the second has $lambda_n$.
$endgroup$
– herb steinberg
Dec 2 '18 at 1:52
$begingroup$
Yes,I'm sorry .
$endgroup$
– mbfkk
Dec 2 '18 at 1:59
$begingroup$
Yes,I'm sorry .
$endgroup$
– mbfkk
Dec 2 '18 at 1:59
$begingroup$
Please correct!
$endgroup$
– herb steinberg
Dec 2 '18 at 3:43
$begingroup$
Please correct!
$endgroup$
– herb steinberg
Dec 2 '18 at 3:43
|
show 1 more comment
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$begingroup$
What is the relationship between m and n?
$endgroup$
– herb steinberg
Dec 2 '18 at 1:41
$begingroup$
They are irrelevant.
$endgroup$
– mbfkk
Dec 2 '18 at 1:46
$begingroup$
Your statement seems to have a typo. The first and third occurrences have $lambda_m$ while the second has $lambda_n$.
$endgroup$
– herb steinberg
Dec 2 '18 at 1:52
$begingroup$
Yes,I'm sorry .
$endgroup$
– mbfkk
Dec 2 '18 at 1:59
$begingroup$
Please correct!
$endgroup$
– herb steinberg
Dec 2 '18 at 3:43