How to find a matrix of linear transformation f : R n → M a t ( n , n ) .Let's say we do f : R 2 → M a t ( 2 , 2 ) given by f ( x , y ) = [ x 2 y x + y x ] We calculate image of canonical basis f ( 1 , 0 ) = [ 1 0 1 1 ] and f ( 0 , 1 ) = [ 0 2 1 0 ] Now the problematic part, ever since when caltulating matrix from vectors R n → R m the approach is to transpose images of standard base ( f ( 1 , 0 , . . . , 0 ) T | f ( 0 , 1 , . . . , 0 ) T | f ( 0 , 0 , . . . , 1 ) T ). We can solve the R → M a t problem by using ( A T | B T | C T . . . ), of course, but is there any way how to shrink the vector so we can succeed something like ( A T | B T | C T . . . )?

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How to find a matrix of linear transformation   f  :      R    n    →  M  a      t          (      n      ,      n      )      .Let&#039;s say we do   f  :      R    2    →  M  a      t          (      2      ,      2      )       given by   f  (  x  ,  y  )  =      [                            x                          2          y                                      x          +          y                          x                      ]  We calculate image of canonical basis   f  (  1  ,  0  )  =      [                            1                          0                                      1                          1                      ]   and   f  (  0  ,  1  )  =      [                            0                          2                                      1                          0                      ]  Now the problematic part, ever since when caltulating matrix from vectors       R    n    →      R    m   the approach is to transpose images of standard base   (  f  (  1  ,  0  ,  .  .  .  ,  0      )          T            |    f  (  0  ,  1  ,  .  .  .  ,  0      )          T            |    f  (  0  ,  0  ,  .  .  .  ,  1      )          T        ). We can solve the   R  →  M  a  t problem by using   (      A    T        |        B    T        |        C    T    .  .  .  ), of course, but is there any way how to shrink the vector so we can succeed something like   (      A    T        |        B    T        |        C    T    .  .  .  )?

trabadero2l · Accepted Answer

f  is a linear transformation from a vector space of dimension   2 to vector space of dimension   4, so its matrix will be of the order   4  ×  2. Now the basis of the space of Matrices of order   2  ×  2 is   {      e          11        ,      e          12        ,      e          21        ,      e          22        }} where       e          i      j        =  1 at the   i      j          th      th entry and   0 at all other places. So   f  (  1  ,  0  )  =      e          11        +      e          21        +      e          22       and   f  (  0  ,  1  )  =  2      e          12        +      e          21       so from here your matrix will be      [                            1                          0                                      0                          2                                      1                          1                                      1                          0                      ]    .

How to find a matrix of linear transformation f:R^n→Mat^(n,n).

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