Answer: The required matrix is
[tex]A=\left[\begin{array}{ccc}6&4&0\\0&6&8\\0&0&6\end{array}\right] .[/tex]
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
[tex]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.[/tex]
Therefore, the matrix A is given by
[tex]A=\left[\begin{array}{ccc}6&4&0\\0&6&8\\0&0&6\end{array}\right] .[/tex]
Thus, the required matrix is
[tex]A=\left[\begin{array}{ccc}6&4&0\\0&6&8\\0&0&6\end{array}\right] .[/tex]
