Can three vectors $ v_1,v_2,v_3 in mathbb R^2$ may be chosen from ${{ u_1,u_2,u_3,u_4}}$ such that $ u=...












2












$begingroup$



Let $u_1,u_2,u_3,u_4$ be vectors in $mathbb{ R^2}$ and
$$ u= sum_{j=1}^{4} t_ju_j;text{ }t_j>0 text{ and } sum_{j=1}^{4} t_j=1$$



Then three vectors $ v_1,v_2,v_3 in mathbb{R}^2$ may be chosen from ${{ u_1,u_2,u_3,u_4}}$ such that
$$ u= sum_{j=1}^{3}s_jv_j ;text{ } s_jgeq 0 text{ and } sum_{j=1}^{3} s_j=1$$




Since $u_1,u_2,u_3,u_4$ are linearly dependent, i can replace $u_4$ (assuming $u_4$ is dependent one) by linear combination of $u_1,u_2,u_3$ so that $ u= sum_{j=1}^{3}s_jv_j$ but I can't claim $sum_{j=1}^{3} s_j=1$ .i'm stuck



Please give me a hint! (Using linear algebra )
I didn't studied topology yet, so it's hard for me to understand topological proof!
Thanks.






linear-algebra vector-spaces linear-transformations







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you mean chosen from or constructed from? Maybe "chosen from the span of.."?
    $endgroup$
    – Paul
    Nov 30 '18 at 12:47












  • $begingroup$
    @Paul chosen from given vectors. And not from span
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:52










  • $begingroup$
    Source : part A 6th question. univ.tifr.res.in/gs2019/Files/GS2012_QP_MTH.pdf
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:53










  • $begingroup$
    @CloudJR you can't do what you say you can. For example, if $;u_1=u_2=0;,;;u_3=(1,0);,;;u_4=(0,1);$ , there in $;u=u_4;$ you can't dispose of $;u_4;$...unless some other conditions are given.
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:04












  • $begingroup$
    @DonAntonio, well i actually assume u4 is dependent, let me edit it thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:45
















2












$begingroup$



Let $u_1,u_2,u_3,u_4$ be vectors in $mathbb{ R^2}$ and
$$ u= sum_{j=1}^{4} t_ju_j;text{ }t_j>0 text{ and } sum_{j=1}^{4} t_j=1$$



Then three vectors $ v_1,v_2,v_3 in mathbb{R}^2$ may be chosen from ${{ u_1,u_2,u_3,u_4}}$ such that
$$ u= sum_{j=1}^{3}s_jv_j ;text{ } s_jgeq 0 text{ and } sum_{j=1}^{3} s_j=1$$




Since $u_1,u_2,u_3,u_4$ are linearly dependent, i can replace $u_4$ (assuming $u_4$ is dependent one) by linear combination of $u_1,u_2,u_3$ so that $ u= sum_{j=1}^{3}s_jv_j$ but I can't claim $sum_{j=1}^{3} s_j=1$ .i'm stuck



Please give me a hint! (Using linear algebra )
I didn't studied topology yet, so it's hard for me to understand topological proof!
Thanks.






linear-algebra vector-spaces linear-transformations







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you mean chosen from or constructed from? Maybe "chosen from the span of.."?
    $endgroup$
    – Paul
    Nov 30 '18 at 12:47












  • $begingroup$
    @Paul chosen from given vectors. And not from span
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:52










  • $begingroup$
    Source : part A 6th question. univ.tifr.res.in/gs2019/Files/GS2012_QP_MTH.pdf
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:53










  • $begingroup$
    @CloudJR you can't do what you say you can. For example, if $;u_1=u_2=0;,;;u_3=(1,0);,;;u_4=(0,1);$ , there in $;u=u_4;$ you can't dispose of $;u_4;$...unless some other conditions are given.
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:04












  • $begingroup$
    @DonAntonio, well i actually assume u4 is dependent, let me edit it thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:45














2












2








2





$begingroup$



Let $u_1,u_2,u_3,u_4$ be vectors in $mathbb{ R^2}$ and
$$ u= sum_{j=1}^{4} t_ju_j;text{ }t_j>0 text{ and } sum_{j=1}^{4} t_j=1$$



Then three vectors $ v_1,v_2,v_3 in mathbb{R}^2$ may be chosen from ${{ u_1,u_2,u_3,u_4}}$ such that
$$ u= sum_{j=1}^{3}s_jv_j ;text{ } s_jgeq 0 text{ and } sum_{j=1}^{3} s_j=1$$




Since $u_1,u_2,u_3,u_4$ are linearly dependent, i can replace $u_4$ (assuming $u_4$ is dependent one) by linear combination of $u_1,u_2,u_3$ so that $ u= sum_{j=1}^{3}s_jv_j$ but I can't claim $sum_{j=1}^{3} s_j=1$ .i'm stuck



Please give me a hint! (Using linear algebra )
I didn't studied topology yet, so it's hard for me to understand topological proof!
Thanks.






linear-algebra vector-spaces linear-transformations







share|cite|improve this question











$endgroup$





Let $u_1,u_2,u_3,u_4$ be vectors in $mathbb{ R^2}$ and
$$ u= sum_{j=1}^{4} t_ju_j;text{ }t_j>0 text{ and } sum_{j=1}^{4} t_j=1$$



Then three vectors $ v_1,v_2,v_3 in mathbb{R}^2$ may be chosen from ${{ u_1,u_2,u_3,u_4}}$ such that
$$ u= sum_{j=1}^{3}s_jv_j ;text{ } s_jgeq 0 text{ and } sum_{j=1}^{3} s_j=1$$




Since $u_1,u_2,u_3,u_4$ are linearly dependent, i can replace $u_4$ (assuming $u_4$ is dependent one) by linear combination of $u_1,u_2,u_3$ so that $ u= sum_{j=1}^{3}s_jv_j$ but I can't claim $sum_{j=1}^{3} s_j=1$ .i'm stuck



Please give me a hint! (Using linear algebra )
I didn't studied topology yet, so it's hard for me to understand topological proof!
Thanks.






linear-algebra vector-spaces linear-transformations




linear-algebra vector-spaces linear-transformations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 30 '18 at 13:52







Cloud JR

















asked Nov 30 '18 at 12:44









Cloud JRCloud JR

882517




882517












  • $begingroup$
    Do you mean chosen from or constructed from? Maybe "chosen from the span of.."?
    $endgroup$
    – Paul
    Nov 30 '18 at 12:47












  • $begingroup$
    @Paul chosen from given vectors. And not from span
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:52










  • $begingroup$
    Source : part A 6th question. univ.tifr.res.in/gs2019/Files/GS2012_QP_MTH.pdf
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:53










  • $begingroup$
    @CloudJR you can't do what you say you can. For example, if $;u_1=u_2=0;,;;u_3=(1,0);,;;u_4=(0,1);$ , there in $;u=u_4;$ you can't dispose of $;u_4;$...unless some other conditions are given.
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:04












  • $begingroup$
    @DonAntonio, well i actually assume u4 is dependent, let me edit it thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:45


















  • $begingroup$
    Do you mean chosen from or constructed from? Maybe "chosen from the span of.."?
    $endgroup$
    – Paul
    Nov 30 '18 at 12:47












  • $begingroup$
    @Paul chosen from given vectors. And not from span
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:52










  • $begingroup$
    Source : part A 6th question. univ.tifr.res.in/gs2019/Files/GS2012_QP_MTH.pdf
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 12:53










  • $begingroup$
    @CloudJR you can't do what you say you can. For example, if $;u_1=u_2=0;,;;u_3=(1,0);,;;u_4=(0,1);$ , there in $;u=u_4;$ you can't dispose of $;u_4;$...unless some other conditions are given.
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:04












  • $begingroup$
    @DonAntonio, well i actually assume u4 is dependent, let me edit it thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:45
















$begingroup$
Do you mean chosen from or constructed from? Maybe "chosen from the span of.."?
$endgroup$
– Paul
Nov 30 '18 at 12:47






$begingroup$
Do you mean chosen from or constructed from? Maybe "chosen from the span of.."?
$endgroup$
– Paul
Nov 30 '18 at 12:47














$begingroup$
@Paul chosen from given vectors. And not from span
$endgroup$
– Cloud JR
Nov 30 '18 at 12:52




$begingroup$
@Paul chosen from given vectors. And not from span
$endgroup$
– Cloud JR
Nov 30 '18 at 12:52












$begingroup$
Source : part A 6th question. univ.tifr.res.in/gs2019/Files/GS2012_QP_MTH.pdf
$endgroup$
– Cloud JR
Nov 30 '18 at 12:53




$begingroup$
Source : part A 6th question. univ.tifr.res.in/gs2019/Files/GS2012_QP_MTH.pdf
$endgroup$
– Cloud JR
Nov 30 '18 at 12:53












$begingroup$
@CloudJR you can't do what you say you can. For example, if $;u_1=u_2=0;,;;u_3=(1,0);,;;u_4=(0,1);$ , there in $;u=u_4;$ you can't dispose of $;u_4;$...unless some other conditions are given.
$endgroup$
– DonAntonio
Nov 30 '18 at 13:04






$begingroup$
@CloudJR you can't do what you say you can. For example, if $;u_1=u_2=0;,;;u_3=(1,0);,;;u_4=(0,1);$ , there in $;u=u_4;$ you can't dispose of $;u_4;$...unless some other conditions are given.
$endgroup$
– DonAntonio
Nov 30 '18 at 13:04














$begingroup$
@DonAntonio, well i actually assume u4 is dependent, let me edit it thanks
$endgroup$
– Cloud JR
Nov 30 '18 at 13:45




$begingroup$
@DonAntonio, well i actually assume u4 is dependent, let me edit it thanks
$endgroup$
– Cloud JR
Nov 30 '18 at 13:45










1 Answer
1






active

oldest

votes


















2












$begingroup$

By its definition, $u$ lies in the convex hull of the $u_i$. It suffices to show that for any $u$ there are three $u_i$ whose convex hull contains $u$.




  • If there are four points on the convex hull, without loss of generality take them to be $u_1u_2u_3u_4$ in that order. Then $u_1u_2u_3$ and $u_3u_4u_1$ partition the hull, so $u$ lies in at least one of them; the points of the enclosing triangle may be taken as the $v_i$.

  • If there are only three points on the convex hull, a correct choice of $v_i$ is simply those three points.

  • The two- and one-point cases are trivial.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I didn't studied convex hull before. Your proof looks geometrically intuitive, but it's hard for me to understand
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:50










  • $begingroup$
    @CloudJR Perhaps you studied "barycentric coordinates" and stuff, which is just next to convex hulls and etc. ?
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:52










  • $begingroup$
    But i will try to understand, and i will comment if i understood or having any problem. Please help ! Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:54










  • $begingroup$
    @DonAntonio nope. I'm not that good at geometry , i am just a second year undergraduate. I think it was a linear algebra problem, but when i tried no linear algebraic tools works... So i ask help here.I'm gonna check it out , in wiki or some books and let you know if i understood. Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:59










  • $begingroup$
    As far as I remember this is standard 2nd. undergraduate year stuff, and many times it is taught within linear algebra and not until topology...
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 14:08











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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

By its definition, $u$ lies in the convex hull of the $u_i$. It suffices to show that for any $u$ there are three $u_i$ whose convex hull contains $u$.




  • If there are four points on the convex hull, without loss of generality take them to be $u_1u_2u_3u_4$ in that order. Then $u_1u_2u_3$ and $u_3u_4u_1$ partition the hull, so $u$ lies in at least one of them; the points of the enclosing triangle may be taken as the $v_i$.

  • If there are only three points on the convex hull, a correct choice of $v_i$ is simply those three points.

  • The two- and one-point cases are trivial.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I didn't studied convex hull before. Your proof looks geometrically intuitive, but it's hard for me to understand
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:50










  • $begingroup$
    @CloudJR Perhaps you studied "barycentric coordinates" and stuff, which is just next to convex hulls and etc. ?
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:52










  • $begingroup$
    But i will try to understand, and i will comment if i understood or having any problem. Please help ! Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:54










  • $begingroup$
    @DonAntonio nope. I'm not that good at geometry , i am just a second year undergraduate. I think it was a linear algebra problem, but when i tried no linear algebraic tools works... So i ask help here.I'm gonna check it out , in wiki or some books and let you know if i understood. Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:59










  • $begingroup$
    As far as I remember this is standard 2nd. undergraduate year stuff, and many times it is taught within linear algebra and not until topology...
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 14:08
















2












$begingroup$

By its definition, $u$ lies in the convex hull of the $u_i$. It suffices to show that for any $u$ there are three $u_i$ whose convex hull contains $u$.




  • If there are four points on the convex hull, without loss of generality take them to be $u_1u_2u_3u_4$ in that order. Then $u_1u_2u_3$ and $u_3u_4u_1$ partition the hull, so $u$ lies in at least one of them; the points of the enclosing triangle may be taken as the $v_i$.

  • If there are only three points on the convex hull, a correct choice of $v_i$ is simply those three points.

  • The two- and one-point cases are trivial.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I didn't studied convex hull before. Your proof looks geometrically intuitive, but it's hard for me to understand
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:50










  • $begingroup$
    @CloudJR Perhaps you studied "barycentric coordinates" and stuff, which is just next to convex hulls and etc. ?
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:52










  • $begingroup$
    But i will try to understand, and i will comment if i understood or having any problem. Please help ! Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:54










  • $begingroup$
    @DonAntonio nope. I'm not that good at geometry , i am just a second year undergraduate. I think it was a linear algebra problem, but when i tried no linear algebraic tools works... So i ask help here.I'm gonna check it out , in wiki or some books and let you know if i understood. Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:59










  • $begingroup$
    As far as I remember this is standard 2nd. undergraduate year stuff, and many times it is taught within linear algebra and not until topology...
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 14:08














2












2








2





$begingroup$

By its definition, $u$ lies in the convex hull of the $u_i$. It suffices to show that for any $u$ there are three $u_i$ whose convex hull contains $u$.




  • If there are four points on the convex hull, without loss of generality take them to be $u_1u_2u_3u_4$ in that order. Then $u_1u_2u_3$ and $u_3u_4u_1$ partition the hull, so $u$ lies in at least one of them; the points of the enclosing triangle may be taken as the $v_i$.

  • If there are only three points on the convex hull, a correct choice of $v_i$ is simply those three points.

  • The two- and one-point cases are trivial.






share|cite|improve this answer









$endgroup$



By its definition, $u$ lies in the convex hull of the $u_i$. It suffices to show that for any $u$ there are three $u_i$ whose convex hull contains $u$.




  • If there are four points on the convex hull, without loss of generality take them to be $u_1u_2u_3u_4$ in that order. Then $u_1u_2u_3$ and $u_3u_4u_1$ partition the hull, so $u$ lies in at least one of them; the points of the enclosing triangle may be taken as the $v_i$.

  • If there are only three points on the convex hull, a correct choice of $v_i$ is simply those three points.

  • The two- and one-point cases are trivial.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 30 '18 at 13:19









Parcly TaxelParcly Taxel

41.6k137299




41.6k137299












  • $begingroup$
    I didn't studied convex hull before. Your proof looks geometrically intuitive, but it's hard for me to understand
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:50










  • $begingroup$
    @CloudJR Perhaps you studied "barycentric coordinates" and stuff, which is just next to convex hulls and etc. ?
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:52










  • $begingroup$
    But i will try to understand, and i will comment if i understood or having any problem. Please help ! Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:54










  • $begingroup$
    @DonAntonio nope. I'm not that good at geometry , i am just a second year undergraduate. I think it was a linear algebra problem, but when i tried no linear algebraic tools works... So i ask help here.I'm gonna check it out , in wiki or some books and let you know if i understood. Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:59










  • $begingroup$
    As far as I remember this is standard 2nd. undergraduate year stuff, and many times it is taught within linear algebra and not until topology...
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 14:08


















  • $begingroup$
    I didn't studied convex hull before. Your proof looks geometrically intuitive, but it's hard for me to understand
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:50










  • $begingroup$
    @CloudJR Perhaps you studied "barycentric coordinates" and stuff, which is just next to convex hulls and etc. ?
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 13:52










  • $begingroup$
    But i will try to understand, and i will comment if i understood or having any problem. Please help ! Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:54










  • $begingroup$
    @DonAntonio nope. I'm not that good at geometry , i am just a second year undergraduate. I think it was a linear algebra problem, but when i tried no linear algebraic tools works... So i ask help here.I'm gonna check it out , in wiki or some books and let you know if i understood. Thanks
    $endgroup$
    – Cloud JR
    Nov 30 '18 at 13:59










  • $begingroup$
    As far as I remember this is standard 2nd. undergraduate year stuff, and many times it is taught within linear algebra and not until topology...
    $endgroup$
    – DonAntonio
    Nov 30 '18 at 14:08
















$begingroup$
I didn't studied convex hull before. Your proof looks geometrically intuitive, but it's hard for me to understand
$endgroup$
– Cloud JR
Nov 30 '18 at 13:50




$begingroup$
I didn't studied convex hull before. Your proof looks geometrically intuitive, but it's hard for me to understand
$endgroup$
– Cloud JR
Nov 30 '18 at 13:50












$begingroup$
@CloudJR Perhaps you studied "barycentric coordinates" and stuff, which is just next to convex hulls and etc. ?
$endgroup$
– DonAntonio
Nov 30 '18 at 13:52




$begingroup$
@CloudJR Perhaps you studied "barycentric coordinates" and stuff, which is just next to convex hulls and etc. ?
$endgroup$
– DonAntonio
Nov 30 '18 at 13:52












$begingroup$
But i will try to understand, and i will comment if i understood or having any problem. Please help ! Thanks
$endgroup$
– Cloud JR
Nov 30 '18 at 13:54




$begingroup$
But i will try to understand, and i will comment if i understood or having any problem. Please help ! Thanks
$endgroup$
– Cloud JR
Nov 30 '18 at 13:54












$begingroup$
@DonAntonio nope. I'm not that good at geometry , i am just a second year undergraduate. I think it was a linear algebra problem, but when i tried no linear algebraic tools works... So i ask help here.I'm gonna check it out , in wiki or some books and let you know if i understood. Thanks
$endgroup$
– Cloud JR
Nov 30 '18 at 13:59




$begingroup$
@DonAntonio nope. I'm not that good at geometry , i am just a second year undergraduate. I think it was a linear algebra problem, but when i tried no linear algebraic tools works... So i ask help here.I'm gonna check it out , in wiki or some books and let you know if i understood. Thanks
$endgroup$
– Cloud JR
Nov 30 '18 at 13:59












$begingroup$
As far as I remember this is standard 2nd. undergraduate year stuff, and many times it is taught within linear algebra and not until topology...
$endgroup$
– DonAntonio
Nov 30 '18 at 14:08




$begingroup$
As far as I remember this is standard 2nd. undergraduate year stuff, and many times it is taught within linear algebra and not until topology...
$endgroup$
– DonAntonio
Nov 30 '18 at 14:08


















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