Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?

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getrdone07tl · Accepted Answer

Let   T be an invertible linear transformation from an   n-dimensional vector space to another,   A its   (  n  ×  n  ) matrix,       B    =  {      α    1    ,  …  ,      α    n    } a basis for the domain,             B        ′    =  {      β    1    ,  …  ,      β    n    } a basis for the co-domain, and   β  =      y    1        β    1    +  ⋯  +      y    n        β    n   a vector in the co-domain.The inverse of   T  ,      T          −      1        ,, is related to       A          −      1       by             [              T                  −          1                    β      ]                      B              =      A          −      1        [  β      ]                            B                ′              ,, where   [  ⋅      ]                  B             denotes the coordinate matrix relative to the ordered basis       B    . Then with       A          i      j              −      1       representing the   i  j entry of       A          −      1        ,, we compute                                          [                          T                              −                1                                      β            ]                                              B                                                          =                  A                      −            1                          [        β                  ]                                                    B                            ′                                                                                      =                  A                      −            1                          [                  y          1                          β          1                +        ⋯        +                  y          n                          β          n                          ]                                                    B                            ′                                                                                      =                  A                      −            1                                    [                                                                      y                  1                                                                                    ⋮                                                                                      y                  n                                                              ]                                                                =                  [                                                                      ∑                                      i                    =                    1                                    n                                                  A                                      1                    i                                                        −                    1                                                                    y                  i                                                                                    ⋮                                                                                      ∑                                      i                    =                    1                                    n                                                  A                                      n                    i                                                        −                    1                                                                    y                  i                                                              ]                ;                                                    T                      −            1                          β                            =                  α          1                          ∑                      i            =            1                    n                          A                      1            i                                −            1                                    y          i                +        ⋯        +                  α          n                          ∑                      i            =            1                    n                          A                      n            i                                −            1                                    y          i                .            If you are satisfied having your inverse in terms of       B    ,, we are done.If, however, you want the inverse in terms of the standard ordered basis, express each       α    i   in terms of the standard basis, and group the terms in the last expression above by the elements of the standard basis.

Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?

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