If T ∈ End ( V ) such that T ( x 1 ) = 2 x 1 + x 2 and T ( x 2 ) = x 1 , and y 1 = 4 x 1 + 2 x 2 and y 2 = x 1 − x 2 , determine the matrix T with respect to the basis { x 1 , x 2 } and with respect to the new basis { y 1 , y 2 }. Furthermore, it is possible to find an invertible matrix P such that P − 1 A P = B, where A is the matrix transformation with respect to the basis { x 1 , x 2 } and B is the matrix transformation with respect to the basis { y 1 , y 2 }.For the first part, I know I need to find some matrix D = C − 1 A C , such that A is the transformation matrix with respect to the standard basis, and C is the change of basis matrix, but I am unsure how to construct C and thus C − 1 . The transformation matrix for T is: A = [ 2 1 1 0 ]

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If   T  ∈  End  (  V  ) such that   T  (      x    1    )  =  2      x    1    +      x    2   and   T  (      x    2    )  =      x    1  , and       y    1    =  4      x    1    +  2      x    2   and       y    2    =      x    1    −      x    2   , determine the matrix   T with respect to the basis   {      x    1    ,      x    2    } and with respect to the new basis   {      y    1    ,      y    2    }. Furthermore, it is possible to find an invertible matrix   P such that       P          −      1        A  P  =  B, where   A is the matrix transformation with respect to the basis   {      x    1    ,      x    2    } and   B is the matrix transformation with respect to the basis   {      y    1    ,      y    2    }.For the first part, I know I need to find some matrix   D  =      C          −      1        A  C , such that   A is the transformation matrix with respect to the standard basis, and   C is the change of basis matrix, but I am unsure how to construct   C and thus       C          −      1      . The transformation matrix for   T is:  A  =      [                            2                          1                                      1                          0                      ]

Maximo Sweeney · Accepted Answer

The matrix of   T in basis   (      x    1    ,      x    2    ) has for columns the coordinates of   T  (      x    1    ) and   T  (      x    2    ) in basis   (      x    1    ,      x    2    ). Thus  A  =      [                            2                          1                                      1                          0                      ]      (no standard basis in this)  .For the basis   (      y    1    ,      y    2    ), use the change of basis matrix from basis   (      x    1    ,      x    2    ) to basis   (      y    1    ,      y    2    ): its column vectoes are the coordinates of       y    1   and       y    2   in basis   (      x    1    ,      x    2    ), i.e.  P  =      [                            4                          1                                      2                          −          1                      ]    .The change of basis formula asserts that the matrix of   T in the new basis is  B  =      P          −      1        A  P  ,    so     A  =  P  B      P          −      1        .All that remains to do is calculating the inverse of   P.

If T ∈ <mtext>End</mtext> ( V ) such that T ( x 1 <

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