If A = [ 1 − 1 2 − 2 1 − 1 1 2 3 ] is the matrix representation of a linear transformation T : P 2 ( x ) → P 2 ( x )with respect to the bases { 1 − x , x ( 1 − x ) , x ( 1 + x ) } and { 1 , 1 + x , 1 + x 2 } then find T. What is the procedure to solve it?

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If  A  =      [                            1                          −          1                          2                                      −          2                          1                          −          1                                      1                          2                          3                      ]  is the matrix representation of a linear transformation  T  :      P    2    (  x  )  →      P    2    (  x  )with respect to the bases   {  1  −  x  ,  x  (  1  −  x  )  ,  x  (  1  +  x  )  } and   {  1  ,  1  +  x  ,  1  +      x    2    } then find T. What is the procedure to solve it?

crociandomh · Accepted Answer

Recall the the matrix of a transformation has as its columns the images of the domain basis vectors expressed relative to a basis of the codomain. Assuming that the first of the given bases is for the domain, this means that  T  [  1  −  x  ]  =  1  ⋅  (  1  )  −  2  ⋅  (  1  +  x  )  +  1  ⋅  (  1  +      x    2    )  =      x    2    −  2  x  .The other two columns give you   T  [  x  (  1  −  x  )  ] and   T  [  x  (  1  +  x  )  ], respectively. Now, express the general second-degree polynomial   a  +  b  x  +  c      x    2   as a linear combination of the domain basis polynomials, i.e., as   α  (  1  −  x  )  +  β  x  (  1  −  x  )  +  γ  x  (  1  +  x  ) and use linearity of   T:  T  [  a  +  b  x  +  c  y  ]  =  α  T  [  1  −  x  ]  +  β  T  [  x  (  1  −  x  )  ]  +  γ  T  [  x  (  1  +  x  )  ]  .  T  [  a  +  b  x  +  c  y  ]  =  α  T  [  1  −  x  ]  +  β  T  [  x  (  1  −  x  )  ]  +  γ  T  [  x  (  1  +  x  )  ]  .

Eden Solomon · Accepted Answer

The columns of the matrix of a transformation are the images of the basis vectors of the domain expressed relative to the basis of the codomain, i.e., it specifies a linear combination of the codomain basis vectors. So, the first column of A tells us that   T  [  1  −  x  ]  =  1  ⋅  (  1  )  −  2  ⋅  (  1  +  x  )  +  1  ⋅  (  1  +      x    2    )  =      x    2    −  2  x, and so on. From that, you should be able to work out what T does to the general polynomial   a  +  b  x  +  c      x    2  . Alternatively, you might convert A to the standard basis and read the solution from that.

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