Show that a translated subspace, i.e., $D={Ev+d mid v in mathbb{R}^p} subseteq mathbb{R}^n$ is a convex set....
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Show that a translated subspace, i.e., $D={Ev+d mid v in mathbb{R}^p} subseteq mathbb{R}^n$ is a convex set where $E in mathbb{R}^{n times p}$ and $d in mathbb{R}^n$.
convex-analysis
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
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closed as off-topic by Randall, DRF, Cesareo, José Carlos Santos, Rebellos Dec 3 '18 at 14:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Randall, DRF, Cesareo, José Carlos Santos, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Show that a translated subspace, i.e., $D={Ev+d mid v in mathbb{R}^p} subseteq mathbb{R}^n$ is a convex set where $E in mathbb{R}^{n times p}$ and $d in mathbb{R}^n$.
convex-analysis
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
$endgroup$
closed as off-topic by Randall, DRF, Cesareo, José Carlos Santos, Rebellos Dec 3 '18 at 14:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Randall, DRF, Cesareo, José Carlos Santos, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
1
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What have you tried? Have you written down the definition of a convex set and tried to check if $D$ satisfies the conditions?
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– angryavian
Dec 3 '18 at 4:37
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Yes. I have considered $x_1$ and $x_2$ in $D$. $x_1=Ev_1+d$ and $x_2=Ev_2+d$. Multiply by $lambda$ and sum. I am not sure if I can write $x_1=Ev_1+d$ and $x_2=Ev_2+d$ or not?
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– Saeed
Dec 3 '18 at 4:55
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Yes, your approach is good.
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– angryavian
Dec 3 '18 at 4:56
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@Saeed please add the information from the comment to the question. Try and explain what you are doing and where you are stuck.
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– DRF
Dec 3 '18 at 9:44
add a comment |
$begingroup$
Show that a translated subspace, i.e., $D={Ev+d mid v in mathbb{R}^p} subseteq mathbb{R}^n$ is a convex set where $E in mathbb{R}^{n times p}$ and $d in mathbb{R}^n$.
convex-analysis
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
$endgroup$
Show that a translated subspace, i.e., $D={Ev+d mid v in mathbb{R}^p} subseteq mathbb{R}^n$ is a convex set where $E in mathbb{R}^{n times p}$ and $d in mathbb{R}^n$.
convex-analysis
convex-analysis
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
asked Dec 3 '18 at 4:24
SaeedSaeed
969310
969310
closed as off-topic by Randall, DRF, Cesareo, José Carlos Santos, Rebellos Dec 3 '18 at 14:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Randall, DRF, Cesareo, José Carlos Santos, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Randall, DRF, Cesareo, José Carlos Santos, Rebellos Dec 3 '18 at 14:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Randall, DRF, Cesareo, José Carlos Santos, Rebellos
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
What have you tried? Have you written down the definition of a convex set and tried to check if $D$ satisfies the conditions?
$endgroup$
– angryavian
Dec 3 '18 at 4:37
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Yes. I have considered $x_1$ and $x_2$ in $D$. $x_1=Ev_1+d$ and $x_2=Ev_2+d$. Multiply by $lambda$ and sum. I am not sure if I can write $x_1=Ev_1+d$ and $x_2=Ev_2+d$ or not?
$endgroup$
– Saeed
Dec 3 '18 at 4:55
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Yes, your approach is good.
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– angryavian
Dec 3 '18 at 4:56
$begingroup$
@Saeed please add the information from the comment to the question. Try and explain what you are doing and where you are stuck.
$endgroup$
– DRF
Dec 3 '18 at 9:44
add a comment |
1
$begingroup$
What have you tried? Have you written down the definition of a convex set and tried to check if $D$ satisfies the conditions?
$endgroup$
– angryavian
Dec 3 '18 at 4:37
$begingroup$
Yes. I have considered $x_1$ and $x_2$ in $D$. $x_1=Ev_1+d$ and $x_2=Ev_2+d$. Multiply by $lambda$ and sum. I am not sure if I can write $x_1=Ev_1+d$ and $x_2=Ev_2+d$ or not?
$endgroup$
– Saeed
Dec 3 '18 at 4:55
$begingroup$
Yes, your approach is good.
$endgroup$
– angryavian
Dec 3 '18 at 4:56
$begingroup$
@Saeed please add the information from the comment to the question. Try and explain what you are doing and where you are stuck.
$endgroup$
– DRF
Dec 3 '18 at 9:44
1
1
$begingroup$
What have you tried? Have you written down the definition of a convex set and tried to check if $D$ satisfies the conditions?
$endgroup$
– angryavian
Dec 3 '18 at 4:37
$begingroup$
What have you tried? Have you written down the definition of a convex set and tried to check if $D$ satisfies the conditions?
$endgroup$
– angryavian
Dec 3 '18 at 4:37
$begingroup$
Yes. I have considered $x_1$ and $x_2$ in $D$. $x_1=Ev_1+d$ and $x_2=Ev_2+d$. Multiply by $lambda$ and sum. I am not sure if I can write $x_1=Ev_1+d$ and $x_2=Ev_2+d$ or not?
$endgroup$
– Saeed
Dec 3 '18 at 4:55
$begingroup$
Yes. I have considered $x_1$ and $x_2$ in $D$. $x_1=Ev_1+d$ and $x_2=Ev_2+d$. Multiply by $lambda$ and sum. I am not sure if I can write $x_1=Ev_1+d$ and $x_2=Ev_2+d$ or not?
$endgroup$
– Saeed
Dec 3 '18 at 4:55
$begingroup$
Yes, your approach is good.
$endgroup$
– angryavian
Dec 3 '18 at 4:56
$begingroup$
Yes, your approach is good.
$endgroup$
– angryavian
Dec 3 '18 at 4:56
$begingroup$
@Saeed please add the information from the comment to the question. Try and explain what you are doing and where you are stuck.
$endgroup$
– DRF
Dec 3 '18 at 9:44
$begingroup$
@Saeed please add the information from the comment to the question. Try and explain what you are doing and where you are stuck.
$endgroup$
– DRF
Dec 3 '18 at 9:44
add a comment |
1 Answer
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If $x_1,x_2 in D$, then $x_1 = Ev_1+d, x_2 = Ev_2+d$ for some $v_1,v_2 in mathbb{R}^p$, respectively. Then, $D$ is convex if $$x_3 = lambda x_1 + (1-lambda)x_2 in D text{ for all }lambdain(0,1)$$ i.e. there exists a $v_3inmathbb{R}^p$ such that $x_3 = Ev_3+d$. Substituting in the expressions for $x_1,x_2$, we see that
$$ lambda(Ev_1+d) + (1-lambda)(Ev_2+d) = Elambda v_1 + lambda d + E(1-lambda)v_2 + (1-lambda)d = E(lambda v_1 + (1-lambda)v_2) + d $$
Since $lambda v_1 + (1-lambda)v_2 in mathbb{R}^p$ for any $v_1,v_2inmathbb{R}^p$ and $lambdain(0,1)subsetmathbb{R}$, we let $v_3 = lambda v_1 + (1-lambda)v_2$, and so it follows that $D$ is convex.
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If $x_1,x_2 in D$, then $x_1 = Ev_1+d, x_2 = Ev_2+d$ for some $v_1,v_2 in mathbb{R}^p$, respectively. Then, $D$ is convex if $$x_3 = lambda x_1 + (1-lambda)x_2 in D text{ for all }lambdain(0,1)$$ i.e. there exists a $v_3inmathbb{R}^p$ such that $x_3 = Ev_3+d$. Substituting in the expressions for $x_1,x_2$, we see that
$$ lambda(Ev_1+d) + (1-lambda)(Ev_2+d) = Elambda v_1 + lambda d + E(1-lambda)v_2 + (1-lambda)d = E(lambda v_1 + (1-lambda)v_2) + d $$
Since $lambda v_1 + (1-lambda)v_2 in mathbb{R}^p$ for any $v_1,v_2inmathbb{R}^p$ and $lambdain(0,1)subsetmathbb{R}$, we let $v_3 = lambda v_1 + (1-lambda)v_2$, and so it follows that $D$ is convex.
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
$endgroup$
add a comment |
$begingroup$
If $x_1,x_2 in D$, then $x_1 = Ev_1+d, x_2 = Ev_2+d$ for some $v_1,v_2 in mathbb{R}^p$, respectively. Then, $D$ is convex if $$x_3 = lambda x_1 + (1-lambda)x_2 in D text{ for all }lambdain(0,1)$$ i.e. there exists a $v_3inmathbb{R}^p$ such that $x_3 = Ev_3+d$. Substituting in the expressions for $x_1,x_2$, we see that
$$ lambda(Ev_1+d) + (1-lambda)(Ev_2+d) = Elambda v_1 + lambda d + E(1-lambda)v_2 + (1-lambda)d = E(lambda v_1 + (1-lambda)v_2) + d $$
Since $lambda v_1 + (1-lambda)v_2 in mathbb{R}^p$ for any $v_1,v_2inmathbb{R}^p$ and $lambdain(0,1)subsetmathbb{R}$, we let $v_3 = lambda v_1 + (1-lambda)v_2$, and so it follows that $D$ is convex.
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
$endgroup$
add a comment |
$begingroup$
If $x_1,x_2 in D$, then $x_1 = Ev_1+d, x_2 = Ev_2+d$ for some $v_1,v_2 in mathbb{R}^p$, respectively. Then, $D$ is convex if $$x_3 = lambda x_1 + (1-lambda)x_2 in D text{ for all }lambdain(0,1)$$ i.e. there exists a $v_3inmathbb{R}^p$ such that $x_3 = Ev_3+d$. Substituting in the expressions for $x_1,x_2$, we see that
$$ lambda(Ev_1+d) + (1-lambda)(Ev_2+d) = Elambda v_1 + lambda d + E(1-lambda)v_2 + (1-lambda)d = E(lambda v_1 + (1-lambda)v_2) + d $$
Since $lambda v_1 + (1-lambda)v_2 in mathbb{R}^p$ for any $v_1,v_2inmathbb{R}^p$ and $lambdain(0,1)subsetmathbb{R}$, we let $v_3 = lambda v_1 + (1-lambda)v_2$, and so it follows that $D$ is convex.
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
$endgroup$
If $x_1,x_2 in D$, then $x_1 = Ev_1+d, x_2 = Ev_2+d$ for some $v_1,v_2 in mathbb{R}^p$, respectively. Then, $D$ is convex if $$x_3 = lambda x_1 + (1-lambda)x_2 in D text{ for all }lambdain(0,1)$$ i.e. there exists a $v_3inmathbb{R}^p$ such that $x_3 = Ev_3+d$. Substituting in the expressions for $x_1,x_2$, we see that
$$ lambda(Ev_1+d) + (1-lambda)(Ev_2+d) = Elambda v_1 + lambda d + E(1-lambda)v_2 + (1-lambda)d = E(lambda v_1 + (1-lambda)v_2) + d $$
Since $lambda v_1 + (1-lambda)v_2 in mathbb{R}^p$ for any $v_1,v_2inmathbb{R}^p$ and $lambdain(0,1)subsetmathbb{R}$, we let $v_3 = lambda v_1 + (1-lambda)v_2$, and so it follows that $D$ is convex.
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
answered Dec 3 '18 at 5:05
AlexanderJ93AlexanderJ93
6,148823
6,148823
add a comment |
add a comment |
1
$begingroup$
What have you tried? Have you written down the definition of a convex set and tried to check if $D$ satisfies the conditions?
$endgroup$
– angryavian
Dec 3 '18 at 4:37
$begingroup$
Yes. I have considered $x_1$ and $x_2$ in $D$. $x_1=Ev_1+d$ and $x_2=Ev_2+d$. Multiply by $lambda$ and sum. I am not sure if I can write $x_1=Ev_1+d$ and $x_2=Ev_2+d$ or not?
$endgroup$
– Saeed
Dec 3 '18 at 4:55
$begingroup$
Yes, your approach is good.
$endgroup$
– angryavian
Dec 3 '18 at 4:56
$begingroup$
@Saeed please add the information from the comment to the question. Try and explain what you are doing and where you are stuck.
$endgroup$
– DRF
Dec 3 '18 at 9:44