Let B be the basis P 3 consisting of the Hermite polynomials 1 , 2 t , − 2 + 4 t 2 and − 12 t + 8 t 3 ; and let p ( t ) = 1 + 16 t 2 − 8 t 3 . Find the coordinate vector of p relative to B. [ p ] B = [ − 7 − 6 − 4 − 1 ]

Question

Let B be the basis             P        3   consisting of the Hermite polynomials   1  ,  2  t  ,  −  2  +  4      t    2   and   −  12  t  +  8      t    3  ; and let   p  (  t  )  =  1  +  16      t    2    −  8      t    3  . Find the coordinate vector of p relative to B.  [  p      ]    B    =      [                            −          7                                      −          6                                      −          4                                      −          1                      ]

tezeuszxcL · Accepted Answer

Let   p  (  t  )  =  a  .1  +  b  (  2  t  )  +  c  (  −  2  +  4      t    2    )  +  d  (  −  12  t  +  8      t    3    )    =  (  8  d  )      t    3    +  (  4  c  )      t    2    +  (  2  b  −  12  d  )  t  +  (  a  −  2  c  )    ⇒  −  8      t    3    +  16      t    2    +  1  =  (  8  d  )      t    3    +  (  4  c  )      t    2    +  (  2  b  −  12  d  )  t  +  (  a  −  2  c  )    ⇒      {                            8          d          =          −          8                                      4          c          =          16                                      2          b          −          12          d          =          0                                      a          −          2          c          =          1                          ⇒      {                            d          =          −          1                                      c          =          4                                      b          =          −          6                                      a          =          9                        So   [  p      ]    B    =      [                            9                                      −          6                                      4                                      −          1                      ]

Let B be the basis if PP_3 consisting of the Hermite polynomials 1,2t, -2+4t^2, and -12t+8t^3; and let p(t)=1+16t^2-8t^3. Find the coordinate vector of p relative to B. [p]_B=[-7,6,-4,-1]

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