A linear transformation T : P 2 → P 2 has matrix with respect to S given by: [ T ] ( S ) = [ 1 / 2 − 3 1 / 2 − 1 4 − 1 1 / 2 2 1 / 2 ] How do you find T ( a + b x + c x 2 )?

Question

A linear transformation   T  :      P    2    →      P    2   has matrix with respect to   S given by:  [  T  ]    (  S  )  =      [                            1                  /          2                          −          3                          1                  /          2                                      −          1                          4                          −          1                                      1                  /          2                          2                          1                  /          2                      ]  How do you find   T  (  a  +  b  x  +  c      x    2    )?

Blaine Foster · Accepted Answer

Every polynomial   S of degree   2 s.t.   S  ∈      P    2   can be represented as a vector in three dimensional space:  S  =  a  +  b  x  +  c      x    2        ⟺          [                            a                                      b                                      c                      ]    ⋅      [                            1                                      x                                                  x            2                                ]    ,therefore we can associate   S with a 3D vector  S  ⟷      [                            a                                      b                                      c                      ]    .When you have the polynomial represented as a vector, applying linear transformation   T, given in form of a matrix, is a piece of cake:  [  T  ]    (  S  )  =      [                            1                  /          2                          −          3                          1                  /          2                                      −          1                          4                          −          1                                      1                  /          2                          2                          1                  /          2                      ]        [                            a                                      b                                      c                      ]    =      [                                        1            2                    a          −          3          b          +                      1            2                    c                                      −          a          +          4          b          −          c                                                  1            2                    a          +          2          b          +                      1            2                    c                      ]    ,which will correspond to the polynomial  T      (    a    +    b    x    +    c          x      2        )    =      (          1      2        a    −    3    b    +          1      2        c    )    +      (    −    a    +    4    b    −    c    )    x  +      (          1      2        a    +    2    b    +          1      2        c    )        x    2

A linear transformation T : P 2 </msub> → P

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