Representing a Linear Transformation as a Matrix T ( [ a b c d ] ) = [ d a b c ] I attempted to apply T to the standard basis of the 2 × 2 matrices. However, I ended up with a 4 × 4 matrix (understandable, as there are 4 matrices in that basis). This matrix I came up with was [ 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 ]

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Representing a Linear Transformation as a Matrix  T  (      [                            a                                  b                                      c                                  d                      ]    )  =      [                            d                                  a                                      b                                  c                      ]  I attempted to apply   T to the standard basis of the   2  ×  2 matrices. However, I ended up with a   4  ×  4 matrix (understandable, as there are   4 matrices in that basis). This matrix I came up with was      [                            0                                  0                                  0                                  1                                      1                                  0                                  0                                  0                                      0                                  1                                  0                                  0                                      0                                  0                                  1                                  0                      ]

cestdugateauhcorr · Accepted Answer

Your issue is that you&#039;re still thinking of the   2  ×  2 matrices as matrices. If you&#039;re going to have them be acted upon by transformations, and you need those transformations to be represented by matrices, you need to represent the elements (some   2  ×  2 matrices) of the base vector space (the space of   2  ×  2 matrices) as column vectors.This may seem confusing because it&#039;s no longer clear what space each matrix or vector is associate with. You could attack this problem by adding some notation to make it clear.Let the vector space that the   2  ×  2 matrices act upon be denoted   V. Then let       M          2      ×      2       denote the set of   2  ×  2 matrices associated with linear transformations on   V. Then what you have could be expressed like this:            [                                    a                                b                                                c                                d                              ]        V    ≡            [                                    a                                                b                                                c                                                d                              ]                M              2        ×        2            The subscript says what vector space each matrix or column vector is associated with.Then your answer is merely            [                                    0                                0                                0                                1                                                1                                0                                0                                0                                                0                                1                                0                                0                                                0                                0                                1                                0                              ]                M              2        ×        2            which is correct.This approach of using subscripts to denote what vector space the matrix or column vector is associated with is by no means standard. I do think this is one reason why higher-level treatments of linear algebra sometimes don&#039;t even use matrices: when you need to talk about a set of matrices itself as a vector space, things get a little confusing.

Payton Salazar · Accepted Answer

Let the basis be       e    1    =      E      11    ,      e    2    =      E      12    ,      e    3    =      E      21    ,      e    4    =      E      22  .Then   T      e    1    =      e    2    ,  T      e    2    =      e    3    ,  T      e    3    =      e    4    ,  T      e    4    =      e    1  .So, your representation is correct, assuming your basis matches mine.

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