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Linear coordinate transformations in multivariable calc vs change of basis


Change of basis matrix confusionChanging basis vectors should change coordinates in the space?What coordinate transformations, if any, are matrices not able to represent?Change of basis - Definition, matrix and relation to diagonalizationinvariants of semilinear transformation under change of basisChange of Basis (Linear Algebra)Intuition behind looking at linear transformations as a change in coordinate systems or change in basisIs a Change of Basis Matrix equivalent to the matrix inverse in this case?Intuitions on understanding system of linear equations as linear transformations3blue1brown's visually doing linear transformations composition













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$begingroup$


When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to another. Assuming that one performs a linear change of variables, can these coordinate transformations be thought of as change of basis matrix? That is, can I view $P^-1AP = B$ as taking the area element (columns of $A$) into the area element in the new coord. system (columns of $B$), where $P$ are the linear change of variables. If so, then since det is invariant under this operation, why do we have to worry about a change in area/vol of the area/vol element?



How does change of basis and change of coordinates relate (if at all) to each other and how do you interpret area being invariant in one situation and not the other?










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    $begingroup$


    When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to another. Assuming that one performs a linear change of variables, can these coordinate transformations be thought of as change of basis matrix? That is, can I view $P^-1AP = B$ as taking the area element (columns of $A$) into the area element in the new coord. system (columns of $B$), where $P$ are the linear change of variables. If so, then since det is invariant under this operation, why do we have to worry about a change in area/vol of the area/vol element?



    How does change of basis and change of coordinates relate (if at all) to each other and how do you interpret area being invariant in one situation and not the other?










    share|cite|improve this question







    New contributor




    jeff is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$














      1












      1








      1


      1



      $begingroup$


      When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to another. Assuming that one performs a linear change of variables, can these coordinate transformations be thought of as change of basis matrix? That is, can I view $P^-1AP = B$ as taking the area element (columns of $A$) into the area element in the new coord. system (columns of $B$), where $P$ are the linear change of variables. If so, then since det is invariant under this operation, why do we have to worry about a change in area/vol of the area/vol element?



      How does change of basis and change of coordinates relate (if at all) to each other and how do you interpret area being invariant in one situation and not the other?










      share|cite|improve this question







      New contributor




      jeff is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to another. Assuming that one performs a linear change of variables, can these coordinate transformations be thought of as change of basis matrix? That is, can I view $P^-1AP = B$ as taking the area element (columns of $A$) into the area element in the new coord. system (columns of $B$), where $P$ are the linear change of variables. If so, then since det is invariant under this operation, why do we have to worry about a change in area/vol of the area/vol element?



      How does change of basis and change of coordinates relate (if at all) to each other and how do you interpret area being invariant in one situation and not the other?







      linear-algebra multivariable-calculus linear-transformations determinant jacobian






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      asked Mar 13 at 18:04









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