Question: For set of vectors x 1 , x 2 , x 1 = ( 1 , 3 ) T , x 2 = ( 4 , 6 ) T are in R 2 . Find the matrix of linear transformation T : R 2 → R 3 such that T x 1 = ( − 2 , 2 , − 7 ) T and T x 2 = ( − 2 , − 4 , − 10 ) T My attempt: So I'm guessing there exists a matrix (a transformation matrix) and it must be of order (2,3) for it to give (-2,2,-7) T when multiplied by (1,3) T : [ t 11 t 12 t 21 t 22 t 31 t 32 ] [ 1 3 ] = [ − 2 2 − 7 ] But how am I supposed to find out all those i j variables? Also, using same on second set of vectors give me inconsistent equations: t 11 + 3 t 12 = − 2 4 t 11 + 6 t 12 = − 2 t 21 + 3 t 22 = 2 4 t 21 + 6 t 22 = − 4 t 31 + 3 t 32 = − 7 4 t 31 + 6 t 32 = − 10

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Question: For set of vectors       x    1    ,      x    2  ,       x    1    =  (  1  ,  3      )    T    ,      x    2    =  (  4  ,  6      )    T   are in       R    2  . Find the matrix of linear transformation   T  :      R    2    →      R    3   such that   T      x    1    =  (  −  2  ,  2  ,  −  7      )    T   and   T      x    2    =  (  −  2  ,  −  4  ,  −  10      )    T  My attempt: So I&#039;m guessing there exists a matrix (a transformation matrix) and it must be of order (2,3) for it to give (-2,2,-7)          T   when multiplied by (1,3)          T  :      [                                        t                      11                                                t                      12                                                            t                      21                                                t                      22                                                            t                      31                                                t                      32                                ]         [                            1                                      3                      ]   =       [                            −          2                                      2                                      −          7                      ]  But how am I supposed to find out all those                 i      j       variables? Also, using same on second set of vectors give me inconsistent equations:                              t                  11                +        3                  t                  12                =        −        2                    4                  t                  11                +        6                  t                  12                =        −        2                                      t                  21                +        3                  t                  22                =        2                    4                  t                  21                +        6                  t                  22                =        −        4                                      t                  31                +        3                  t                  32                =        −        7                    4                  t                  31                +        6                  t                  32                =        −        10

barbesdestyle2k · Accepted Answer

It might be easier if you use a different vector. Notice that       x    2    −  2      x    1    =  (  2  ,  0      )    T  . Since you know that   T  (      x    2    −  2      x    1    )  =  T  (      x    2    )  −  2  T  (      x    1    ), you can set up a simpler set of equations. Then try to get a vector   (  0  ,  a      )    T  , for some   a.

Question: For set of vectors x 1 </msub> , x 2 </msub> ,

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