Question

Find the matrix A of the linear transformation T(f(t))= 4f'(t)+6f(t) from P2 T0 P2 with respect to the standard basis of P2 {1,t,t^2}
You don't need to answer the full question i have the answer just tips on where to start and on what to do would be nice.

Answer 1

Answer:  The required matrix is

$A=\left[\begin{array}{ccc}6&4&0\\0&6&8\\0&0&6\end{array}\right] .$

Step-by-step explanation:  The given linear transformation is

T(f(t)) = 4f'(t) + 6f(t).

We are to find the matrix A of T from P² to P² with respect to the standard basis P² = {1, t, t²}.

We have

$T(1)=4\times\dfrac{d}{dt}1+6\times1=0+6=6\times 1+0\times t+0\times t^2,\\\\T(t)=4\times\dfrac{d}{dt}t+6\times t=4+6t=4\times1+6\times t+0\times t^2,\\\\T(t^2)=4\times\dfrac{d}{dt}t^2+6\timest^2=8t+6t^2=0\times 1+8\times t+6\times t^2.$

Therefore, the matrix A is given by

$A=\left[\begin{array}{ccc}6&4&0\\0&6&8\\0&0&6\end{array}\right] .$

Thus, the required matrix is

$A=\left[\begin{array}{ccc}6&4&0\\0&6&8\\0&0&6\end{array}\right] .$