How to do the following change of Basis












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$begingroup$


$V=mathbb{Q}^3$ The two following Bases of V are given



$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;



Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$



Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$



"How would I begin to solve this problem/what steps should be taken to solve it."






linear-algebra change-of-basis







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  • $begingroup$
    Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
    $endgroup$
    – T. Fo
    Dec 14 '18 at 3:40
















1












$begingroup$


$V=mathbb{Q}^3$ The two following Bases of V are given



$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;



Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$



Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$



"How would I begin to solve this problem/what steps should be taken to solve it."






linear-algebra change-of-basis







share|cite|improve this question











$endgroup$












  • $begingroup$
    Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
    $endgroup$
    – T. Fo
    Dec 14 '18 at 3:40














1












1








1





$begingroup$


$V=mathbb{Q}^3$ The two following Bases of V are given



$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;



Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$



Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$



"How would I begin to solve this problem/what steps should be taken to solve it."






linear-algebra change-of-basis







share|cite|improve this question











$endgroup$




$V=mathbb{Q}^3$ The two following Bases of V are given



$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;



Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$



Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$



"How would I begin to solve this problem/what steps should be taken to solve it."






linear-algebra change-of-basis




linear-algebra change-of-basis






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edited Dec 14 '18 at 3:12









Key Flex

8,00261233




8,00261233










asked Dec 14 '18 at 3:05









DanDan

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  • $begingroup$
    Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
    $endgroup$
    – T. Fo
    Dec 14 '18 at 3:40


















  • $begingroup$
    Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
    $endgroup$
    – T. Fo
    Dec 14 '18 at 3:40
















$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40




$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40










2 Answers
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$begingroup$

Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    $gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.



    $gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.






    share|cite|improve this answer









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      2 Answers
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      active

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      2 Answers
      2






      active

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      active

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      0












      $begingroup$

      Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.






      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.






        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.






          share|cite|improve this answer









          $endgroup$



          Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 14 '18 at 3:22









          T. FoT. Fo

          466311




          466311























              0












              $begingroup$

              $gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.



              $gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                $gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.



                $gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  $gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.



                  $gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.






                  share|cite|improve this answer









                  $endgroup$



                  $gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.



                  $gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 14 '18 at 15:18









                  gandalf61gandalf61

                  8,649725




                  8,649725






























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