Let T : R 3 -&gt; R 2 be a linear transformation satisfying: T ( [ 1 , 0 , 1 ] ) = [ 2 , 3 ] , T ( [ 2 , 1 , 3 ] ) = [ − 1 , 0 ] , T ( [ 0 , 0 , 1 ] ) = [ 3 , 7 ]Find T ( [ x , y , z ] )

Question

Let   T  :              R    3   -&amp;gt;         R    2   be a linear transformation satisfying:  T  (  [  1  ,  0  ,  1  ]  )  =  [  2  ,  3  ]  ,           T  (  [  2  ,  1  ,  3  ]  )  =  [  −  1  ,  0  ]  ,           T  (  [  0  ,  0  ,  1  ]  )  =  [  3  ,  7  ]Find         T  (  [  x  ,  y  ,  z  ]  )

Gloletheods6g · Accepted Answer

As it is a linear transformation, you can write   T as a matrix  A  =      (                            a                          b                          c                                      d                          e                          f                      )  Starting from your last condition   T  (  [  0  ,  0  ,  1  ]  )  =  [  3  ,  7  ], meaning that      (                            a                          b                          c                                      d                          e                          f                      )        (                            0                                      0                                      1                      )    =      (                            3                                      7                      )  It means that   c  =  3 and   f  =  7. You can follow with the other two conditions in order to determine the rest of the coefficients in a similar way. Let me know if you have problems to finish it

Alessandra Clarke · Accepted Answer

You can use   A  =      [                            T          (                      e            1                    )                          T          (                      e            2                    )                          T          (                      e            3                    )                      ]   where       e    1    =      [                            1                                      0                                      0                      ]    ,      e    2    =      [                            0                                      1                                      0                      ]    ,      e    3    =      [                            0                                      0                                      1                      ]  , and         e    1    =      v    1    −      v    3   and       e    2    =      v    2    −  2      v    1    −      v    3   where       v    1    =      [                            1                                      0                                      1                      ]    ,      v    2    =      [                            2                                      1                                      3                      ]  , and       v    3    =      e    3  ;so   T  (      e    1    )  =      [                            2                                      3                      ]    −      [                            3                                      7                      ]    =      [                            −          1                                      −          4                      ]   and   T  (      e    2    )  =      [                            −          1                                      0                      ]    −  2      [                            2                                      3                      ]    −      [                            3                                      7                      ]    =      [                            −          8                                      −          13                      ]

Let T : <mtext> </mtext> <mtext> </mtext> <mi mathvariant="double-

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