Linear transformation and its matrixtwo bases: A = { v 1 , v 2 , v 3 } and B = { 2 v 1 , v 2 + v 3 , − v 1 + 2 v 2 − v 3 }There is also a linear transformation: T : R 3 → R 3 Matrix in base A: M T A = [ 1 2 3 4 5 6 1 1 0 ] Now I am to find matrix of the linear transformation T in base B.I have found two transition matrixes (from base A to B and from B to A): P A B = [ 2 0 − 1 0 1 2 0 1 − 1 ] ( P A B ) − 1 = P B A = [ 1 2 1 6 − 1 6 0 1 3 2 3 0 1 3 − 1 3 ] How can I find M T B ?

Question

Linear transformation and its matrixtwo bases:  A  =  {      v    1    ,      v    2    ,      v    3    } and   B  =  {  2      v    1    ,      v    2    +      v    3    ,  −      v    1    +  2      v    2    −      v    3    }There is also a linear transformation:   T  :        R    3    →        R    3  Matrix in base   A:      M      T      A    =      [                            1                          2                          3                                      4                          5                          6                                      1                          1                          0                      ]  Now I am to find matrix of the linear transformation   T in base   B.I have found two transition matrixes (from base   A to   B and from   B to   A):      P      A      B    =      [                            2                          0                          −          1                                      0                          1                          2                                      0                          1                          −          1                      ]    (      P      A      B        )          −      1        =      P      B      A    =      [                                        1            2                                                1            6                                                              −              1                        6                                                0                                      1            3                                                2            3                                                0                                      1            3                                                              −              1                        3                                ]  How can I find       M      T      B  ?

EreneDreaceaw · Accepted Answer

Some notation: for a vector   v  ∈        R    3  , let   [  v      ]    A   denote the coordinate vector of   v with respect to the basis   A, and let   [  v      ]    B   denote the coordinate vector of   v with respect to the basis   B. both of these are column vectors. To put this another way,  [  v      ]    A    =      (                                        a            1                                                            a            2                                                            a            3                                )      ⟺    v  =      a    1        v    1    +      a    2        v    2    +      a    3        v    3  We can think of       M    T    A   as a &quot;machine&quot; with the property that, with the usual matrix multiplication,       M    T    A    [  v      ]    A    =  [  T  (  v  )      ]    A  . Similarly,       P    A    B  satisfies       P    A    B    [  v      ]    B    =  [  v      ]    A  , whereas       P    B    A   satisfies       P    B    A    [  v      ]    A    =  [  v      ]    B  . What we want is to &quot;build&quot; is a machine       M    T    B   for which       M    T    B    [  v      ]    B    =  [  T  (  v  )      ]    B  .We can break the process of going from   [  v      ]    B   to   [  T  (  v  )      ]    B   into three steps, each of which uses machinery that we already have. First, go from   [  v      ]    B   to   [  v      ]    A   with       P    A    B    [  v      ]    B    =  [  v      ]    A  . Then, go from   [  v      ]    A   to   [  T  (  v  )      ]    A   using       M    T    A    [  v      ]    A    =  [  T  (  v  )      ]    A  . Then, go from   [  T  (  v  )      ]    A   to   [  T  (  v  )      ]    B   using       P    B    A    [  T  (  v  )      ]    A    =  [  T  (  v  )      ]    B  .Putting it all together, we have  [  T  (  v  )      ]    B    =      P    B    A    [  T  (  v  )      ]    A    =      P    B    A    (      M    T    A    [  v      ]    A    )  =      P    B    A        M    T    A    (      P    A    B    [  v      ]    B    )  =  (      P    B    A        M    T    A        P    A    B    )  [  v      ]    B  What we have found, then, is that the matrix which takes us from   [  v      ]    B   to   [  T  (  v  )      ]    B   is the product       P    B    A        M    T    A        P    A    B    =  (      P    A    B        )          −      1            M    T    A        P    A    B  . So, this is our matrix       M    T    B  .

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