what is Matrix of a linear transformation?












1












$begingroup$


I am a student and I'm studying linear algebra. in the Sheldon Axler book in the part "The Spectral Theorem" and in this video he mentions the operator $T$ as
$$T=begin{pmatrix}2&-3\3&2end{pmatrix}$$



then after finding it's eigenvectors $2 + 3i$ and $2 - 3i$ (in the video I've linked), he says: "with respect to this basis, the matrix of $T$ is the diagonal matrix":
$$
begin{pmatrix}
2 + 3i&0 \
0&2 - 3i
end{pmatrix}
$$



I am confused. $T$ already mentions as $T=begin{pmatrix}2&-3\3&2end{pmatrix}$ so the matrix of $T$ must be $begin{pmatrix}2&-3\3&2end{pmatrix}$ so I think that I don't know the meaning of the matrix of $T$.



could you help me figure it out?






linear-algebra matrices operator-theory spectral-theory







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    I am a student and I'm studying linear algebra. in the Sheldon Axler book in the part "The Spectral Theorem" and in this video he mentions the operator $T$ as
    $$T=begin{pmatrix}2&-3\3&2end{pmatrix}$$



    then after finding it's eigenvectors $2 + 3i$ and $2 - 3i$ (in the video I've linked), he says: "with respect to this basis, the matrix of $T$ is the diagonal matrix":
    $$
    begin{pmatrix}
    2 + 3i&0 \
    0&2 - 3i
    end{pmatrix}
    $$



    I am confused. $T$ already mentions as $T=begin{pmatrix}2&-3\3&2end{pmatrix}$ so the matrix of $T$ must be $begin{pmatrix}2&-3\3&2end{pmatrix}$ so I think that I don't know the meaning of the matrix of $T$.



    could you help me figure it out?






    linear-algebra matrices operator-theory spectral-theory







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I am a student and I'm studying linear algebra. in the Sheldon Axler book in the part "The Spectral Theorem" and in this video he mentions the operator $T$ as
      $$T=begin{pmatrix}2&-3\3&2end{pmatrix}$$



      then after finding it's eigenvectors $2 + 3i$ and $2 - 3i$ (in the video I've linked), he says: "with respect to this basis, the matrix of $T$ is the diagonal matrix":
      $$
      begin{pmatrix}
      2 + 3i&0 \
      0&2 - 3i
      end{pmatrix}
      $$



      I am confused. $T$ already mentions as $T=begin{pmatrix}2&-3\3&2end{pmatrix}$ so the matrix of $T$ must be $begin{pmatrix}2&-3\3&2end{pmatrix}$ so I think that I don't know the meaning of the matrix of $T$.



      could you help me figure it out?






      linear-algebra matrices operator-theory spectral-theory







      share|cite|improve this question











      $endgroup$




      I am a student and I'm studying linear algebra. in the Sheldon Axler book in the part "The Spectral Theorem" and in this video he mentions the operator $T$ as
      $$T=begin{pmatrix}2&-3\3&2end{pmatrix}$$



      then after finding it's eigenvectors $2 + 3i$ and $2 - 3i$ (in the video I've linked), he says: "with respect to this basis, the matrix of $T$ is the diagonal matrix":
      $$
      begin{pmatrix}
      2 + 3i&0 \
      0&2 - 3i
      end{pmatrix}
      $$



      I am confused. $T$ already mentions as $T=begin{pmatrix}2&-3\3&2end{pmatrix}$ so the matrix of $T$ must be $begin{pmatrix}2&-3\3&2end{pmatrix}$ so I think that I don't know the meaning of the matrix of $T$.



      could you help me figure it out?






      linear-algebra matrices operator-theory spectral-theory




      linear-algebra matrices operator-theory spectral-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 5 at 22:52







      Peyman mohseni kiasari

















      asked Jan 5 at 22:31









      Peyman mohseni kiasariPeyman mohseni kiasari

      13711




      13711






















          1 Answer
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          3












          $begingroup$

          Matrices only define linear transformations relative to some basis. They don't describe a linear transformation on their own. Thus implicit in $$T=begin{pmatrix} 2&-3 \ 3 & 2 end{pmatrix}$$
          is the statement that the vector space has some basis already and that this is the matrix for $T$ with respect to that basis. E.g., perhaps it's $k^2$, so it has the standard basis vectors $begin{pmatrix} 1 \ 0end{pmatrix}$ and $begin{pmatrix} 0 \ 1end{pmatrix}$ already, and $T$ has that matrix with respect to that basis.



          If $V$ is a vector space, then a linear transformation $T:Vto V$ is not a matrix, but rather a function with nice properties that respect the vector space structure. We can then describe it using bases and a matrix, but that's only a description, and the description depends on the basis used to compute the matrix.



          As for how to compute the matrix of a linear transformation with respect to some basis, look here for some random notes I found online, or in any decent linear algebra textbook, like presumably in Axler's book somewhere (It appears to be section 3C). For an example of how to do this in a particular case, you can look at this question.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            great answer, thanks.
            $endgroup$
            – Peyman mohseni kiasari
            Jan 5 at 22:46










          • $begingroup$
            @Peyman Glad it was helpful, also the matrix for a linear transformation appears to be defined in section 3C of Axler's book based on a table of contents I found online.
            $endgroup$
            – jgon
            Jan 5 at 22:48












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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3












          $begingroup$

          Matrices only define linear transformations relative to some basis. They don't describe a linear transformation on their own. Thus implicit in $$T=begin{pmatrix} 2&-3 \ 3 & 2 end{pmatrix}$$
          is the statement that the vector space has some basis already and that this is the matrix for $T$ with respect to that basis. E.g., perhaps it's $k^2$, so it has the standard basis vectors $begin{pmatrix} 1 \ 0end{pmatrix}$ and $begin{pmatrix} 0 \ 1end{pmatrix}$ already, and $T$ has that matrix with respect to that basis.



          If $V$ is a vector space, then a linear transformation $T:Vto V$ is not a matrix, but rather a function with nice properties that respect the vector space structure. We can then describe it using bases and a matrix, but that's only a description, and the description depends on the basis used to compute the matrix.



          As for how to compute the matrix of a linear transformation with respect to some basis, look here for some random notes I found online, or in any decent linear algebra textbook, like presumably in Axler's book somewhere (It appears to be section 3C). For an example of how to do this in a particular case, you can look at this question.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            great answer, thanks.
            $endgroup$
            – Peyman mohseni kiasari
            Jan 5 at 22:46










          • $begingroup$
            @Peyman Glad it was helpful, also the matrix for a linear transformation appears to be defined in section 3C of Axler's book based on a table of contents I found online.
            $endgroup$
            – jgon
            Jan 5 at 22:48
















          3












          $begingroup$

          Matrices only define linear transformations relative to some basis. They don't describe a linear transformation on their own. Thus implicit in $$T=begin{pmatrix} 2&-3 \ 3 & 2 end{pmatrix}$$
          is the statement that the vector space has some basis already and that this is the matrix for $T$ with respect to that basis. E.g., perhaps it's $k^2$, so it has the standard basis vectors $begin{pmatrix} 1 \ 0end{pmatrix}$ and $begin{pmatrix} 0 \ 1end{pmatrix}$ already, and $T$ has that matrix with respect to that basis.



          If $V$ is a vector space, then a linear transformation $T:Vto V$ is not a matrix, but rather a function with nice properties that respect the vector space structure. We can then describe it using bases and a matrix, but that's only a description, and the description depends on the basis used to compute the matrix.



          As for how to compute the matrix of a linear transformation with respect to some basis, look here for some random notes I found online, or in any decent linear algebra textbook, like presumably in Axler's book somewhere (It appears to be section 3C). For an example of how to do this in a particular case, you can look at this question.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            great answer, thanks.
            $endgroup$
            – Peyman mohseni kiasari
            Jan 5 at 22:46










          • $begingroup$
            @Peyman Glad it was helpful, also the matrix for a linear transformation appears to be defined in section 3C of Axler's book based on a table of contents I found online.
            $endgroup$
            – jgon
            Jan 5 at 22:48














          3












          3








          3





          $begingroup$

          Matrices only define linear transformations relative to some basis. They don't describe a linear transformation on their own. Thus implicit in $$T=begin{pmatrix} 2&-3 \ 3 & 2 end{pmatrix}$$
          is the statement that the vector space has some basis already and that this is the matrix for $T$ with respect to that basis. E.g., perhaps it's $k^2$, so it has the standard basis vectors $begin{pmatrix} 1 \ 0end{pmatrix}$ and $begin{pmatrix} 0 \ 1end{pmatrix}$ already, and $T$ has that matrix with respect to that basis.



          If $V$ is a vector space, then a linear transformation $T:Vto V$ is not a matrix, but rather a function with nice properties that respect the vector space structure. We can then describe it using bases and a matrix, but that's only a description, and the description depends on the basis used to compute the matrix.



          As for how to compute the matrix of a linear transformation with respect to some basis, look here for some random notes I found online, or in any decent linear algebra textbook, like presumably in Axler's book somewhere (It appears to be section 3C). For an example of how to do this in a particular case, you can look at this question.






          share|cite|improve this answer











          $endgroup$



          Matrices only define linear transformations relative to some basis. They don't describe a linear transformation on their own. Thus implicit in $$T=begin{pmatrix} 2&-3 \ 3 & 2 end{pmatrix}$$
          is the statement that the vector space has some basis already and that this is the matrix for $T$ with respect to that basis. E.g., perhaps it's $k^2$, so it has the standard basis vectors $begin{pmatrix} 1 \ 0end{pmatrix}$ and $begin{pmatrix} 0 \ 1end{pmatrix}$ already, and $T$ has that matrix with respect to that basis.



          If $V$ is a vector space, then a linear transformation $T:Vto V$ is not a matrix, but rather a function with nice properties that respect the vector space structure. We can then describe it using bases and a matrix, but that's only a description, and the description depends on the basis used to compute the matrix.



          As for how to compute the matrix of a linear transformation with respect to some basis, look here for some random notes I found online, or in any decent linear algebra textbook, like presumably in Axler's book somewhere (It appears to be section 3C). For an example of how to do this in a particular case, you can look at this question.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 5 at 22:48

























          answered Jan 5 at 22:42









          jgonjgon

          16.4k32143




          16.4k32143












          • $begingroup$
            great answer, thanks.
            $endgroup$
            – Peyman mohseni kiasari
            Jan 5 at 22:46










          • $begingroup$
            @Peyman Glad it was helpful, also the matrix for a linear transformation appears to be defined in section 3C of Axler's book based on a table of contents I found online.
            $endgroup$
            – jgon
            Jan 5 at 22:48


















          • $begingroup$
            great answer, thanks.
            $endgroup$
            – Peyman mohseni kiasari
            Jan 5 at 22:46










          • $begingroup$
            @Peyman Glad it was helpful, also the matrix for a linear transformation appears to be defined in section 3C of Axler's book based on a table of contents I found online.
            $endgroup$
            – jgon
            Jan 5 at 22:48
















          $begingroup$
          great answer, thanks.
          $endgroup$
          – Peyman mohseni kiasari
          Jan 5 at 22:46




          $begingroup$
          great answer, thanks.
          $endgroup$
          – Peyman mohseni kiasari
          Jan 5 at 22:46












          $begingroup$
          @Peyman Glad it was helpful, also the matrix for a linear transformation appears to be defined in section 3C of Axler's book based on a table of contents I found online.
          $endgroup$
          – jgon
          Jan 5 at 22:48




          $begingroup$
          @Peyman Glad it was helpful, also the matrix for a linear transformation appears to be defined in section 3C of Axler's book based on a table of contents I found online.
          $endgroup$
          – jgon
          Jan 5 at 22:48


















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