Find the matrix A Find the matrix A of the linear transforma

Find the matrix A of the linear transformation T(f(t)) = 6f\'(t)+ 2f(t) from P2 to P2 with respect to the standard basis for P2, {1,t, t^2).

Solution

we know that

[1]\' = 0,

=> T(t=1) = 6*f \'(t) + 2*f(t) = 6*f \'(0) + 2*f(0) = 6*0 + 2*1 = 2

=> 2*1 + 0*t + 0*t^2.

So the first column of the matrix A has entries 2, 0, 0.

Since [t]\' = 1, you have

T(t=t) = 6*f \'(t) + 2*f(t) = 6*f \'(t) + 2*f(t) = 6*1 + 2*t = 6 + 2t

=> 6*1 + 2*t + 0*t^2

so the second column of A has the entries 6 , 2 , 0.

Since [t^2]\' = 2t, you have

T(t^2) = 6*f \'(t) + 2*f(t) = 6*f \'(t) + 2*f(t) = 6*2t + 2*t^2 = 12t + 2t^2
=> 0*1 + 12t + 2t^2

So the third column of the matrix A has entries 0, 12 , 2.

hence the matrix A = | 2 0   0 |

                               | 6 2   0 |

                               | 0 12 2 |