Let V denote the real vector space R 2 and ψ : V → V be a real linear transformation such that ψ ( ( 1 , 0 ) ) = ( 11 , 8 ) and ψ ( ( 0 , 1 ) ) = ( 4 , 3 ). Express the image ψ ( ( x , y ) ) of ( x , y ) in terms of x and y. Assume that w 1 = ( 4 , 5 ) and w 2 = ( 9 , 11 ) form an ordered basis B for V . Working from the denition determine the matrix M B B ( ψ ) with respect to the basis B.does M B B ( ψ ) = [ − 469 − 1048 388 867 ] ?

Question

Let   V denote the real vector space       R    2   and   ψ  :  V  →  V be a real linear transformation such that   ψ  (  (  1  ,  0  )  )  =  (  11  ,  8  ) and   ψ  (  (  0  ,  1  )  )  =  (  4  ,  3  ). Express the image   ψ  (  (  x  ,  y  )  ) of   (  x  ,  y  ) in terms of   x and   y. Assume that       w    1    =  (  4  ,  5  ) and       w    2    =  (  9  ,  11  ) form an ordered basis   B for   V . Working from the denition determine the matrix       M    B    B    (  ψ  ) with respect to the basis   B.does       M    B    B    (  ψ  )  =      [                            −          469                          −          1048                                      388                          867                      ]  ?

Shayna Woods · Accepted Answer

enote   E the standard basis (1,0), (0,1). By definition, the matrix       M    E    E    (  ψ  ) (using your notation) is given by      M    E    E    (  ψ  )  =      [                            11                          4                                      8                          3                      ]    .Now, the transition matrix from   B to   E is given by      M    B    E    (  i  d  )  =      [                            4                          9                                      5                          11                      ]    .Hence the transition matrix from   E to   B is given by      M    E    B    (  i  d  )  =            [                                    4                                9                                                5                                11                              ]              −      1        =      1          −      1            [                            11                          −          9                                      −          5                          4                      ]    =      [                            −          11                          9                                      5                          −          4                      ]    .Combining all these, the matrix       M    B    B    (  ψ  ) is given by       M    B    B    (  ψ  )  =      M    E    B    (  i  d  )  ⋅      M    E    E    (  ψ  )  ⋅      M    B    E    (  i  d  )  =      [                            −          11                          9                                      5                          −          4                      ]    ⋅      [                            11                          4                                      8                          3                      ]    ⋅      [                            4                          9                                      5                          11                      ]    =  .  .  .

Mary Ashley · Accepted Answer

The answer to the first part of the question is implicit, but I&#039;ll bring it out explicitly.  (  x  ,  y  )  =  x  (  1  ,  0  )  +  y  (  0  ,  1  ), and   ψ  is linear, so  ψ  (  x  ,  y  )  =  x  ψ  (  1  ,  0  )  +  y  ψ  (  0  ,  1  )  =  x  (  11  ,  8  )  +  y  (  4  ,  3  )  =  (  11  x  +  4  y  ,  8  x  +  3  y  )

Let V denote the real vector space R 2 </msup> and ψ : V

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