We have a linear transformation T : M 2 × 2 ( F ) → F by T ( A ) = t r ( A ).We want to compute the matrix representation [ T ] from α to γ coordinates. M 2 × 2 has the standard ordered basis " α" for a 2 × 2 matrix. F has the standard basis " γ" for a scalar.My understanding is that any matrix A in the space M 2 × 2 can be represented by the linear combination: a α 1 + b α 2 + c α 3 + d α 4 and the t r ( A ) can be written as: ( a + c ) γ .I'm not sure how to get the matrix representation from this.

Question

We have a linear transformation   T  :      M          2      ×      2        (  F  )  →  F by   T  (  A  )  =  t  r  (  A  ).We want to compute the matrix representation   [  T  ] from   α to   γ coordinates.      M          2      ×      2       has the standard ordered basis &quot;  α&quot; for a   2  ×  2 matrix.   F has the standard basis &quot;  γ&quot; for a scalar.My understanding is that any matrix   A in the space       M          2      ×      2       can be represented by the linear combination:  a      α    1    +  b      α    2    +  c      α    3    +  d      α    4  and the   t  r  (  A  ) can be written as:  (  a  +  c  )  γ  .I&#039;m not sure how to get the matrix representation from this.

podesect · Accepted Answer

Our linear map is                    T                    :                                      M                      2            ×            2                          (        F        )                    →                    F                                        A                    ↦                    tr        ⁡        (        A        )            The basis   α for         M          2      ×      2        (  F  ) is  α  =      {          [                                    1                                0                                                0                                0                              ]        ,          [                                    0                                1                                                0                                0                              ]        ,          [                                    0                                0                                                1                                0                              ]              [                                    0                                0                                                0                                1                              ]        }  and the basis   γ for   F is   γ  =  {  1  }.Now, note that                    T                  (                      [                                                            1                                                  0                                                                              0                                                  0                                                      ]                    )                            =                    1                            T                  (                      [                                                            0                                                  1                                                                              0                                                  0                                                      ]                    )                            =                    0                            T                  (                      [                                                            0                                                  0                                                                              1                                                  0                                                      ]                    )                            =                    0                            T                  (                      [                                                            0                                                  0                                                                              0                                                  1                                                      ]                    )                            =                    1            This implies that   [  T      ]    α    γ   is the   1  ×  4 matrix  [  T      ]    α    γ    =      [                            1                          0                          0                          1                      ]  This allows us to interpret trace as matrix multiplication. Note that  A  =      [                            a                          b                                      c                          d                      ]    =  a      [                            1                          0                                      0                          0                      ]    +  b      [                            0                          1                                      0                          0                      ]    +  c      [                            0                          0                                      1                          0                      ]    +  d      [                            0                          0                                      0                          1                      ]  which means that relative to   α the matrix   A can be viewed as the vector      [                            a                                      b                                      c                                      d                      ]  Hence  tr  ⁡  (  A  )  =      [                            1                          0                          0                          1                      ]        [                            a                                      b                                      c                                      d                      ]    =  a  +  d

We have a linear transformation T : M 2 × 2 ( F ) →...

Answered question

Answer & Explanation