In the vectorspace R 2 two vectors are given (Which also form the basis a): a 1 = ( 8 , − 3 ) and a 2 = ( 5 , − 2 )A linear mapping is determined by: f ( a 1 ) = 2 a 1 − 4 a 2 and f ( a 2 ) = − a 1 + 2 a 2 How can one determine the transformation matrix of f with respect to the standard e-basis?

Question

In the vectorspace         R    2   two vectors are given (Which also form the basis a):       a    1    =  (  8  ,  −  3  ) and       a    2    =  (  5  ,  −  2  )A linear mapping is determined by:   f  (      a    1    )  =  2      a    1    −  4      a    2   and   f  (      a    2    )  =  −      a    1    +  2      a    2   How can one determine the transformation matrix of f with respect to the standard e-basis?

radcas87gex5r · Accepted Answer

Given a linear map   T  :  V  &amp;#x2192;  W suppose   A  =  {      a    1    ,  .  .  .  ,      a    n    } is a basis for   V and   B  =  {      b    1    ,  .  .  .  ,      b    m    } is a basis for   W i.e they are finite dimensional vector spaces. We will write   T  =      (                            T          (                      a            1                    )                          &amp;#x22EF;                          T          (                      a            n                    )                      )  .Given a vector       v    &amp;#x2192;  , you would like to find   [  T      v    &amp;#x2192;        ]    B  . First we have       v    &amp;#x2192;    =      &amp;#x2211;    k        v    k        a    k   which we identify with   (      v    1    ,  .  .  .  ,      v    n        )    T    :=  [  v      ]    A   i.e its coordinates representation. And now, by the definition of matrix multiplication,                    T                        v          &amp;#x2192;                =                  &amp;#x2211;                      j            =            1                    n                T        (                  a          j                )        &amp;#xA0;                  v          j                =                  (                                                    T                (                                  a                  1                                )                                            &amp;#x22EF;                                            T                (                                  a                  n                                )                                              )                          (                                                                      v                  1                                                                                    &amp;#x22EE;                                                                                      v                  n                                                              )                &amp;#x21D2;        [        T                        v          &amp;#x2192;                          ]          B                                    =                  &amp;#x2211;                      j            =            1                    n                [        T        (                  a          j                )                  ]          B                &amp;#xA0;                  v          j                                                                =                                                    (                                                                    [                    T                    (                                          a                      1                                        )                                          ]                      B                                                                            &amp;#x22EF;                                                        [                    T                    (                                          a                      n                                        )                                          ]                      B                                                                                  )                        &amp;#x23DF;                                        :=                          T                              B                A                                                    [        v                  ]          A                    Therefore, given the linear map   f  :  (        R    2    ,  {      a    1    ,      a    2    }  :=  A  )  &amp;#x2192;  (        R    2    ,  {  (  1  ,  0      )    T    ,  (  0  ,  1      )    T    }  :=  B  ), we have that the coordinate transformation matrix is given by,      f          B      A        =      (                            [          f          (                      a            1                    )                      ]            B                                    [          f          (                      a            2                    )                      ]            B                                )  i.e we have to write   f  (      a    1    )  =  2      a    1    &amp;#x2212;  4      a    2   and   f  (      a    2    )  =  &amp;#x2212;      a    1    +  2      a    2   in terms of the unit basis vectors   (  1  ,  0      )    T   and   (  0  ,  1      )    T  . Given       a    1    =  (  8  ,  &amp;#x2212;  3      )    T    ,      a    2    =  (  5  ,  &amp;#x2212;  2      )    T   then   f  (      a    1    )  =  (  &amp;#x2212;  4  ,  2      )    T   and   f  (      a    2    )  =  (  2  ,  &amp;#x2212;  1      )    T  . It follows that the matrix is,      f          B      A        =      (                                        (                                                            &amp;#x2212;                  4                                                                              2                                                      )                                                (                                                            2                                                                              &amp;#x2212;                  1                                                      )                                )  If you want the transformation matrix with respect to basis   A then you are trying to compute       f          A      A      . It follows that           f              A        A              =          I              A        B              &amp;#x2218;          f              B        A             where,      I          A      B        =      (                            2                          5                                      &amp;#x2212;          3                          &amp;#x2212;          8                      )  i.e we have,      f          A      A        =      (                            2                          &amp;#x2212;          1                                      &amp;#x2212;          4                          2                      )

In the vectorspace <mi mathvariant="double-struck">R 2 </msup> two vectors are

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