Let V be a 3 dimensional vector space over a field F and fix ( v 1 , v 2 , v 3 ) as a basis. Consider a linear transformation T : V → V. Then we have T ( v 1 ) = a 11 v 1 + a 21 v 2 + a 31 v 3 T ( v 2 ) = a 12 v 1 + a 22 v 2 + a 32 v 3 T ( v 3 ) = a 13 v 1 + a 23 v 2 + a 33 v 3 So that we can identify T by the matrix ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) But then when I read several linear algebra book, it said if T ( v i ) = ∑ j a i j v j , then we can identify T by the matrix ( a i j ). My problem is: isn't the matrix is ( a j i ) instead of ( a i j )?

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Let V  be a 3 dimensional vector space over a field F and fix   (      v    1    ,      v    2    ,      v    3    ) as a basis. Consider a linear transformation   T  :  V  →  V. Then we have  T  (      v    1    )  =      a      11        v    1    +      a      21        v    2    +      a      31        v    3    T  (      v    2    )  =      a      12        v    1    +      a      22        v    2    +      a      32        v    3    T  (      v    3    )  =      a      13        v    1    +      a      23        v    2    +      a      33        v    3  So that we can identify T by the matrix      (                                        a                      11                                                a                      12                                                a                      13                                                            a                      21                                                a                      22                                                a                      23                                                            a                      31                                                a                      32                                                a                      33                                )  But then when I read several linear algebra book, it said if   T  (      v    i    )  =      ∑    j        a          i      j            v    j   , then we can identify T by the matrix   (      a          i      j        ). My problem is: isn&#039;t the matrix is   (      a          j      i        ) instead of   (      a          i      j        )?

kawiarkahh · Accepted Answer

Your book has a typo (as did I); it should be  T  (        v    i    )  =      ∑    j        a          j      i              v    j    .Here is an example: in the basis   {        v    1    ,        v    2    ,        v    3    }, we know that         v    1   corresponds to the column vector      (                            1                                      0                                      0                      )    .Applying the algorithm for matrix multiplication,      (                                        a                      11                                                a                      12                                                a                      13                                                            a                      21                                                a                      22                                                a                      23                                                            a                      31                                                a                      32                                                a                      33                                )        (                            1                                      0                                      0                      )    =      (                                        a                      11                                                                      a                      21                                                                      a                      31                                          )    =      a      11          v    1    +      a      21          v    2    +      a      31          v    3    =      ∑          j      =      1        3        a          j      1              v    j    .

Patatiniuh · Accepted Answer

They&#039;re written it wrong. They should&#039;ve written   T  (      v    j    )  =      ∑    i        a          i      j            v    i   - edited so that this is correct now. I saw that i wasn&#039;t free in the original way it was written but not the transpose part. I think a form that doesn&#039;t transpose is inherently easier to work with.

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