Mixing
“minimization” problem in Euclidean space related to orthonormal basis
$begingroup$
I have stumbled upon an exercice for second year undegraduate student majoring in economics which I find quite demanding. I have an idea for the solution, but it seems awfully complicated, and I am wondering if there was a simpler way to solve it. This goes as follows:
Let $(E, langle, rangle)$ be a euclidean space of dimension $n geq 2$.
1) For any $x in E$, show that $||x|| = sqrt{n}$ if and only if there exists $(e_1, dots, e_n)$ an orthonormal basis of $E$ such that $x =e_1+ ldots + e_n$.
2) For any $(x,y) in E$, show that $||x|| = sqrt{n}$, $||y|| = sqrt{frac{n(n+1)(2n+1)}{6}}$ and $langle x,y rangle = frac{n(n+1)}{2}$ if and only if there eixsts $(e_1, ldots, e_n)$ an orthonormal basis of $E$ such that $x = e_1 + ldots + e_n$ and $y = e_1 + 2e_2 + ldots + ne_n$
The first question can be solved relatively simply : one proceeds inductively. The induction step goes as follows : find a vector $e_{t_0}$ such that $langle x, e_{t_0} rangle = 1$ and $||e_{t_0}|| = 1$. Then, consider $x' = x-e_{t_0}$. It is easily checked that $x'$ is orthogonal to $e_{t_0}$ and that $x'$ satifies the induction hypothesis. To find $e_{t_0}$ use the intermediate value theorem with the hypotheses $langle x, x/||x|| rangle = sqrt{n} >1$ and $langle x, z rangle = 0$ for any $z$ orthogonal to $x$.
For the second question, one could proceed also inductively. It would be sufficient to find $e_{t_0} in E$ such that $langle x,e_{t_0} rangle = 1$, $langle y, e_{t_0} rangle = n$ and $||e_{t_0}|| = 1$. I have a vague idea on how to find such a $e_{t_0}$ but it seems horribly complicated and certainly not acessible to second year undergraduate student majoring in economics. I am wondering if there was a (realtively) simple way to find the solution for this second question?
linear-algebra optimization orthonormal
asked Jan 9 at 10:34
LibliLibli
1164
$endgroup$
add a comment |
$begingroup$
I have stumbled upon an exercice for second year undegraduate student majoring in economics which I find quite demanding. I have an idea for the solution, but it seems awfully complicated, and I am wondering if there was a simpler way to solve it. This goes as follows:
Let $(E, langle, rangle)$ be a euclidean space of dimension $n geq 2$.
1) For any $x in E$, show that $||x|| = sqrt{n}$ if and only if there exists $(e_1, dots, e_n)$ an orthonormal basis of $E$ such that $x =e_1+ ldots + e_n$.
2) For any $(x,y) in E$, show that $||x|| = sqrt{n}$, $||y|| = sqrt{frac{n(n+1)(2n+1)}{6}}$ and $langle x,y rangle = frac{n(n+1)}{2}$ if and only if there eixsts $(e_1, ldots, e_n)$ an orthonormal basis of $E$ such that $x = e_1 + ldots + e_n$ and $y = e_1 + 2e_2 + ldots + ne_n$
The first question can be solved relatively simply : one proceeds inductively. The induction step goes as follows : find a vector $e_{t_0}$ such that $langle x, e_{t_0} rangle = 1$ and $||e_{t_0}|| = 1$. Then, consider $x' = x-e_{t_0}$. It is easily checked that $x'$ is orthogonal to $e_{t_0}$ and that $x'$ satifies the induction hypothesis. To find $e_{t_0}$ use the intermediate value theorem with the hypotheses $langle x, x/||x|| rangle = sqrt{n} >1$ and $langle x, z rangle = 0$ for any $z$ orthogonal to $x$.
For the second question, one could proceed also inductively. It would be sufficient to find $e_{t_0} in E$ such that $langle x,e_{t_0} rangle = 1$, $langle y, e_{t_0} rangle = n$ and $||e_{t_0}|| = 1$. I have a vague idea on how to find such a $e_{t_0}$ but it seems horribly complicated and certainly not acessible to second year undergraduate student majoring in economics. I am wondering if there was a (realtively) simple way to find the solution for this second question?
linear-algebra optimization orthonormal
asked Jan 9 at 10:34
LibliLibli
1164
$endgroup$
add a comment |
$begingroup$
I have stumbled upon an exercice for second year undegraduate student majoring in economics which I find quite demanding. I have an idea for the solution, but it seems awfully complicated, and I am wondering if there was a simpler way to solve it. This goes as follows:
Let $(E, langle, rangle)$ be a euclidean space of dimension $n geq 2$.
1) For any $x in E$, show that $||x|| = sqrt{n}$ if and only if there exists $(e_1, dots, e_n)$ an orthonormal basis of $E$ such that $x =e_1+ ldots + e_n$.
2) For any $(x,y) in E$, show that $||x|| = sqrt{n}$, $||y|| = sqrt{frac{n(n+1)(2n+1)}{6}}$ and $langle x,y rangle = frac{n(n+1)}{2}$ if and only if there eixsts $(e_1, ldots, e_n)$ an orthonormal basis of $E$ such that $x = e_1 + ldots + e_n$ and $y = e_1 + 2e_2 + ldots + ne_n$
The first question can be solved relatively simply : one proceeds inductively. The induction step goes as follows : find a vector $e_{t_0}$ such that $langle x, e_{t_0} rangle = 1$ and $||e_{t_0}|| = 1$. Then, consider $x' = x-e_{t_0}$. It is easily checked that $x'$ is orthogonal to $e_{t_0}$ and that $x'$ satifies the induction hypothesis. To find $e_{t_0}$ use the intermediate value theorem with the hypotheses $langle x, x/||x|| rangle = sqrt{n} >1$ and $langle x, z rangle = 0$ for any $z$ orthogonal to $x$.
For the second question, one could proceed also inductively. It would be sufficient to find $e_{t_0} in E$ such that $langle x,e_{t_0} rangle = 1$, $langle y, e_{t_0} rangle = n$ and $||e_{t_0}|| = 1$. I have a vague idea on how to find such a $e_{t_0}$ but it seems horribly complicated and certainly not acessible to second year undergraduate student majoring in economics. I am wondering if there was a (realtively) simple way to find the solution for this second question?
linear-algebra optimization orthonormal
asked Jan 9 at 10:34
LibliLibli
1164
$endgroup$
I have stumbled upon an exercice for second year undegraduate student majoring in economics which I find quite demanding. I have an idea for the solution, but it seems awfully complicated, and I am wondering if there was a simpler way to solve it. This goes as follows:
Let $(E, langle, rangle)$ be a euclidean space of dimension $n geq 2$.
1) For any $x in E$, show that $||x|| = sqrt{n}$ if and only if there exists $(e_1, dots, e_n)$ an orthonormal basis of $E$ such that $x =e_1+ ldots + e_n$.
2) For any $(x,y) in E$, show that $||x|| = sqrt{n}$, $||y|| = sqrt{frac{n(n+1)(2n+1)}{6}}$ and $langle x,y rangle = frac{n(n+1)}{2}$ if and only if there eixsts $(e_1, ldots, e_n)$ an orthonormal basis of $E$ such that $x = e_1 + ldots + e_n$ and $y = e_1 + 2e_2 + ldots + ne_n$
The first question can be solved relatively simply : one proceeds inductively. The induction step goes as follows : find a vector $e_{t_0}$ such that $langle x, e_{t_0} rangle = 1$ and $||e_{t_0}|| = 1$. Then, consider $x' = x-e_{t_0}$. It is easily checked that $x'$ is orthogonal to $e_{t_0}$ and that $x'$ satifies the induction hypothesis. To find $e_{t_0}$ use the intermediate value theorem with the hypotheses $langle x, x/||x|| rangle = sqrt{n} >1$ and $langle x, z rangle = 0$ for any $z$ orthogonal to $x$.
For the second question, one could proceed also inductively. It would be sufficient to find $e_{t_0} in E$ such that $langle x,e_{t_0} rangle = 1$, $langle y, e_{t_0} rangle = n$ and $||e_{t_0}|| = 1$. I have a vague idea on how to find such a $e_{t_0}$ but it seems horribly complicated and certainly not acessible to second year undergraduate student majoring in economics. I am wondering if there was a (realtively) simple way to find the solution for this second question?
linear-algebra optimization orthonormal
linear-algebra optimization orthonormal
asked Jan 9 at 10:34
LibliLibli
1164
asked Jan 9 at 10:34
LibliLibli
1164
edited Jan 9 at 11:33
Libli
asked Jan 9 at 10:34
LibliLibli
1164
asked Jan 9 at 10:34
LibliLibli
1164
asked Jan 9 at 10:34
LibliLibli
1164
1164
add a comment |
add a comment |
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