Let B = { u 1 , u 2 , u 3 } with u 1 = 1, u 2 = x, u 3 = x 2 denote a basis for the space of polynomials of the second order polynomials P 2 . Let T ( a 0 + a 1 x + a 2 x 2 ) = a 0 + a 1 ( x − 1 ) + a 2 ( x − 1 ) 2 be a linear transformation form P 2 to itself. Construct the representation matrix [ T ] B B ..

Question

Let   B  =  {      u    1    ,      u    2    ,      u    3    } with       u    1    =  1,       u    2    =  x,       u    3    =      x    2   denote a basis for the space of polynomials of the second order polynomials       P    2  . Let   T  (      a    0    +      a    1    x  +      a    2        x    2    )  =      a    0    +      a    1    (  x  −  1  )  +      a    2    (  x  −  1      )    2   be a linear transformation form       P    2   to itself. Construct the representation matrix   [  T      ]          B      B        ..

verdesett014ci · Accepted Answer

T  (  1  +  0  x  +  0      x    2    )  =  1  +  0  (  x  −  1  )  +  0  (  x  −  1      )    2    =  1So the 1st column of the transformation matrix is       (                            1                                      0                                      0                      )  .  T  (  0  +  1  x  +  0      x    2    )  =  0  +  1  (  x  −  1  )  +  0  (  x  −  1      )    2    =  x  −  1So the 2nd column of the transformation matrix is       (                            −          1                                      1                                      0                      )  .  T  (  0  +  0  x  +  1      x    2    )  =  0  +  0  (  x  −  1  )  +  1  (  x  −  1      )    2    =  (  x  −  1      )    2    =      x    2    −  2  x  +  1So the 3rd column of the transformation matrix is       (                            1                                      −          2                                      1                      )  .So we get that       T          B      B        =      (                            1                          −          1                          1                                      0                          1                          −          2                                      0                          0                          1                      )  .

Let B = <mo fence="false" stretchy="false">{ u 1 </msub> , u

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