Let T : R 2 → R 3 and T ( − 2 , 3 ) = ( − 1 , 0 , 1 ) and T ( 1 , 2 ) = ( 0 , − 1 , 0 )Obtain the canonical matrix of T and the transformation T ( x , y ).

Question

Let   T  :        R    2    →        R    3   and   T  (  −  2  ,  3  )  =  (  −  1  ,  0  ,  1  ) and   T  (  1  ,  2  )  =  (  0  ,  −  1  ,  0  )Obtain the canonical matrix of   T and the transformation   T  (  x  ,  y  ).

Kaylin Barry · Accepted Answer

The canonical basis of the vector space         R    n   is the set of vectors       e      1    ,  …  ,      e      n    ∈        R    n   defined byGiven a linear transformation    T  :        R    n    →        R    m  , the canonical matrix for   T is the   m  ×  n matrix formed by      [                                                                        T          (                      e            1                    )                          T          (                      e            2                    )                          ⋯                          T          (                      e            n                    )                                                                  ]  Thus, for example,                      [                                                                                                                T              (                              e                1                            )                                      T              (                              e                2                            )                                      ⋯                                      T              (                              e                n                            )                                                                                                          ]            ⏟              canonical matrix for       T                            [                                            1                                                          ⋮                                                          0                                      ]            ⏟                e      1        =  T  (      e    1    )Therefore, your task is to use the facts  T      [                            −          2                                      3                      ]    =      [                            −          1                                      0                                      1                      ]        T      [                            1                                      2                      ]    =      [                            0                                      1                                      0                      ]  to compute  T      [                            1                                      0                      ]    =      [                            ?                                      ?                                      ?                      ]        T      [                            0                                      1                      ]    =      [                            ?                                      ?                                      ?                      ]  How to do that? Here is an example. If there existed real numbers   a and   b (hint: they exist!) such that      [                            1                                      0                      ]    =      [                            −          2          a          +          b                                      3          a          +          2          b                      ]    =  a      [                            −          2                                      3                      ]    +  b      [                            1                                      2                      ]  then you could use the fact that   T is a linear transformation to conclude that  T      [                            1                                      0                      ]    =  T      (    a          [                                    −            2                                                3                              ]        +    b          [                                    1                                                2                              ]        )    =  a  ⋅  T      [                            −          2                                      3                      ]    +  b  ⋅  T      [                            1                                      2                      ]    =  a      [                            −          1                                      0                                      1                      ]    +  b      [                            0                                      1                                      0                      ]    =      [                            −          a                                      b                                      a                      ]

Let T : <mi mathvariant="double-struck">R 2 </msup> →<!-- \rightarrow -

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