When is a set of vectors in $mathbb{C}^n$ where the $n$th component of each vector is the $(n-1)$st power of...
1
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Suppose $alpha_1,alpha_2,dots,alpha_n in mathbb{C}$. Consider the set of vectors $S = {xin mathbb{C}^n : x = (1, alpha_i, alpha^2_i, dots,alpha^{n-1}_i), 1leq i leq n }$. Under what conditions on $alpha_1, alpha_2, dots , alpha_n$ is $S$ linearly dependent?
linear-algebra
asked Jan 12 at 16:21
FortoxFortox
5718
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add a comment |
1
$begingroup$
Suppose $alpha_1,alpha_2,dots,alpha_n in mathbb{C}$. Consider the set of vectors $S = {xin mathbb{C}^n : x = (1, alpha_i, alpha^2_i, dots,alpha^{n-1}_i), 1leq i leq n }$. Under what conditions on $alpha_1, alpha_2, dots , alpha_n$ is $S$ linearly dependent?
linear-algebra
asked Jan 12 at 16:21
FortoxFortox
5718
$endgroup$
2
$begingroup$
If and only if $alpha_i = alpha_j$ for some $ine j$. See en.wikipedia.org/wiki/Vandermonde_matrix.
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– Song
Jan 12 at 16:24
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@DavidC.Ullrich Thanks, edited
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– Fortox
Jan 12 at 16:37
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If it's not clear why it matters consider a simpler example: (i) If $ainBbb R$ then ${xinBbb R:xge a}=[a,infty)$. (ii) Otoh ${xinBbb R:ainBbb R. xge a}=Bbb R$. (Because in the second example there's a "such that $ainBbb R$" inside the braces; for any $x$ there is $a$ such that $xge a$.)
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– David C. Ullrich
Jan 12 at 16:48
add a comment |
1
1
1
$begingroup$
Suppose $alpha_1,alpha_2,dots,alpha_n in mathbb{C}$. Consider the set of vectors $S = {xin mathbb{C}^n : x = (1, alpha_i, alpha^2_i, dots,alpha^{n-1}_i), 1leq i leq n }$. Under what conditions on $alpha_1, alpha_2, dots , alpha_n$ is $S$ linearly dependent?
linear-algebra
asked Jan 12 at 16:21
FortoxFortox
5718
$endgroup$
Suppose $alpha_1,alpha_2,dots,alpha_n in mathbb{C}$. Consider the set of vectors $S = {xin mathbb{C}^n : x = (1, alpha_i, alpha^2_i, dots,alpha^{n-1}_i), 1leq i leq n }$. Under what conditions on $alpha_1, alpha_2, dots , alpha_n$ is $S$ linearly dependent?
linear-algebra
linear-algebra
asked Jan 12 at 16:21
FortoxFortox
5718
asked Jan 12 at 16:21
FortoxFortox
5718
edited Jan 12 at 16:36
Fortox
asked Jan 12 at 16:21
FortoxFortox
5718
asked Jan 12 at 16:21
FortoxFortox
5718
asked Jan 12 at 16:21
FortoxFortox
5718
5718
2
$begingroup$
If and only if $alpha_i = alpha_j$ for some $ine j$. See en.wikipedia.org/wiki/Vandermonde_matrix.
$endgroup$
– Song
Jan 12 at 16:24
$begingroup$
@DavidC.Ullrich Thanks, edited
$endgroup$
– Fortox
Jan 12 at 16:37
$begingroup$
If it's not clear why it matters consider a simpler example: (i) If $ainBbb R$ then ${xinBbb R:xge a}=[a,infty)$. (ii) Otoh ${xinBbb R:ainBbb R. xge a}=Bbb R$. (Because in the second example there's a "such that $ainBbb R$" inside the braces; for any $x$ there is $a$ such that $xge a$.)
$endgroup$
– David C. Ullrich
Jan 12 at 16:48
add a comment |
2
$begingroup$
If and only if $alpha_i = alpha_j$ for some $ine j$. See en.wikipedia.org/wiki/Vandermonde_matrix.
$endgroup$
– Song
Jan 12 at 16:24
$begingroup$
@DavidC.Ullrich Thanks, edited
$endgroup$
– Fortox
Jan 12 at 16:37
$begingroup$
If it's not clear why it matters consider a simpler example: (i) If $ainBbb R$ then ${xinBbb R:xge a}=[a,infty)$. (ii) Otoh ${xinBbb R:ainBbb R. xge a}=Bbb R$. (Because in the second example there's a "such that $ainBbb R$" inside the braces; for any $x$ there is $a$ such that $xge a$.)
$endgroup$
– David C. Ullrich
Jan 12 at 16:48
2
2
$begingroup$
If and only if $alpha_i = alpha_j$ for some $ine j$. See en.wikipedia.org/wiki/Vandermonde_matrix.
$endgroup$
– Song
Jan 12 at 16:24
$begingroup$
If and only if $alpha_i = alpha_j$ for some $ine j$. See en.wikipedia.org/wiki/Vandermonde_matrix.
$endgroup$
– Song
Jan 12 at 16:24
$begingroup$
@DavidC.Ullrich Thanks, edited
$endgroup$
– Fortox
Jan 12 at 16:37
$begingroup$
@DavidC.Ullrich Thanks, edited
$endgroup$
– Fortox
Jan 12 at 16:37
$begingroup$
If it's not clear why it matters consider a simpler example: (i) If $ainBbb R$ then ${xinBbb R:xge a}=[a,infty)$. (ii) Otoh ${xinBbb R:ainBbb R. xge a}=Bbb R$. (Because in the second example there's a "such that $ainBbb R$" inside the braces; for any $x$ there is $a$ such that $xge a$.)
$endgroup$
– David C. Ullrich
Jan 12 at 16:48
$begingroup$
If it's not clear why it matters consider a simpler example: (i) If $ainBbb R$ then ${xinBbb R:xge a}=[a,infty)$. (ii) Otoh ${xinBbb R:ainBbb R. xge a}=Bbb R$. (Because in the second example there's a "such that $ainBbb R$" inside the braces; for any $x$ there is $a$ such that $xge a$.)
$endgroup$
– David C. Ullrich
Jan 12 at 16:48
add a comment |
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2
$begingroup$
If and only if $alpha_i = alpha_j$ for some $ine j$. See en.wikipedia.org/wiki/Vandermonde_matrix.
$endgroup$
– Song
Jan 12 at 16:24
$begingroup$
@DavidC.Ullrich Thanks, edited
$endgroup$
– Fortox
Jan 12 at 16:37
$begingroup$
If it's not clear why it matters consider a simpler example: (i) If $ainBbb R$ then ${xinBbb R:xge a}=[a,infty)$. (ii) Otoh ${xinBbb R:ainBbb R. xge a}=Bbb R$. (Because in the second example there's a "such that $ainBbb R$" inside the braces; for any $x$ there is $a$ such that $xge a$.)
$endgroup$
– David C. Ullrich
Jan 12 at 16:48