Finding Coordinate Vector?












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In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?










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    – José Carlos Santos
    Dec 7 '18 at 9:13
















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$begingroup$


In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?










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  • $begingroup$
    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 9:13














0












0








0





$begingroup$


In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?










share|cite|improve this question











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In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?







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edited Dec 7 '18 at 9:12









José Carlos Santos

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asked Dec 7 '18 at 9:05









user9605051user9605051

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  • $begingroup$
    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    $endgroup$
    – José Carlos Santos
    Dec 7 '18 at 9:13


















  • $begingroup$
    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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    – José Carlos Santos
    Dec 7 '18 at 9:13
















$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 7 '18 at 9:13




$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 7 '18 at 9:13










2 Answers
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$begingroup$

It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.






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    $begingroup$

    The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.



    If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then



    $4mathbf{e_1}+3mathbf{e_2}=[2,3]$



    $6mathbf{e_1}+6mathbf{e_2}=[4,5]$



    $Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$



    $Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$



    $ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$






    share|cite|improve this answer









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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

      oldest

      votes









      active

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      votes






      active

      oldest

      votes









      3












      $begingroup$

      It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.






          share|cite|improve this answer









          $endgroup$



          It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 7 '18 at 9:07









          Kavi Rama MurthyKavi Rama Murthy

          53.5k32055




          53.5k32055























              0












              $begingroup$

              The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.



              If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then



              $4mathbf{e_1}+3mathbf{e_2}=[2,3]$



              $6mathbf{e_1}+6mathbf{e_2}=[4,5]$



              $Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$



              $Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$



              $ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.



                If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then



                $4mathbf{e_1}+3mathbf{e_2}=[2,3]$



                $6mathbf{e_1}+6mathbf{e_2}=[4,5]$



                $Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$



                $Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$



                $ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.



                  If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then



                  $4mathbf{e_1}+3mathbf{e_2}=[2,3]$



                  $6mathbf{e_1}+6mathbf{e_2}=[4,5]$



                  $Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$



                  $Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$



                  $ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$






                  share|cite|improve this answer









                  $endgroup$



                  The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.



                  If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then



                  $4mathbf{e_1}+3mathbf{e_2}=[2,3]$



                  $6mathbf{e_1}+6mathbf{e_2}=[4,5]$



                  $Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$



                  $Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$



                  $ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 7 '18 at 9:30









                  gandalf61gandalf61

                  7,986625




                  7,986625






























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