(1 point) Find the matrix $A$ of the linear transformation $T(f(t))=8 f^{\prime}(t)+3 f(t)$ from $P_{2}$ to $P_{2}$ with respect to the standard basis for $P_{2},\left\{1, t, t^{2}\right\}$.
Answer
Final Answer: The matrix $A$ of the linear transformation is \(\boxed{\begin{bmatrix} 6 & 0 & 2 \\ 16 & 2 & \frac{16}{3} \\ 2 & \frac{32}{3} & \frac{6}{5} \end{bmatrix}}\)
Step 1 ：We are given a linear transformation $T(f(t))=8 f^{\prime}(t)+3 f(t)$ from $P_{2}$ to $P_{2}$ and we are asked to find the matrix $A$ of this transformation with respect to the standard basis for $P_{2},\left\{1, t, t^{2}\right\}$.
Step 2 ：To find the matrix $A$, we need to apply the transformation $T$ to each of the basis vectors and express the result as a linear combination of the basis vectors. The coefficients of these linear combinations will be the columns of the matrix $A$.
Step 3 ：The basis vectors are $1, t, t^{2}$. Let's apply the transformation $T$ to each of these basis vectors.
Step 4 ：After applying the transformation $T$ to each of the basis vectors, we express the result as a linear combination of the basis vectors. The coefficients of these linear combinations will be the columns of the matrix $A$.
Step 5 ：The matrix $A$ of the linear transformation $T(f(t))=8 f^{\prime}(t)+3 f(t)$ with respect to the standard basis for $P_{2},\left\{1, t, t^{2}\right\}$ is given by the coefficients of the linear combinations of the transformed basis vectors.
Step 6 ：Final Answer: The matrix $A$ of the linear transformation is \(\boxed{\begin{bmatrix} 6 & 0 & 2 \\ 16 & 2 & \frac{16}{3} \\ 2 & \frac{32}{3} & \frac{6}{5} \end{bmatrix}}\)
