To find the matrix representation of the linear transformation $T$ relative to the basis $B = \{ 1, t, t^2 \}$, we need to express the transformation $T$ of each basis element in terms of the basis $\{ 1, t, t^2 \}$.

Step-by-step solution:

1. Linear transformation $T$:

   We are given that the transformation $T$ is defined as: $T(a_0 + a_1 t + a_2 t^2) = 3a_0 + (5a_0 - 2a_1) t + (4a_1 + a_2) t^2$ This is a linear map, and we will apply it to each basis element $1, t, t^2$ to find the matrix representation of $T$.

2. Apply $T$ to each basis element:

   Let's apply the transformation to the elements of the basis $B = \{1, t, t^2\}$.

   • For $T(1)$, set $a_0 = 1$, $a_1 = 0$, and $a_2 = 0$ in the transformation: $T(1) = 3(1) + (5(1) - 2(0)) t + (4(0) + 0) t^2 = 3 + 5t$ So, $T(1) = 3 + 5t$, which is written as $3 \cdot 1 + 5 \cdot t + 0 \cdot t^2$.

   • For $T(t)$, set $a_0 = 0$, $a_1 = 1$, and $a_2 = 0$: $T(t) = 3(0) + (5(0) - 2(1)) t + (4(1) + 0) t^2 = -2t + 4t^2$ So, $T(t) = -2t + 4t^2$, which is written as $0 \cdot 1 + (-2) \cdot t + 4 \cdot t^2$.

   • For $T(t^2)$, set $a_0 = 0$, $a_1 = 0$, and $a_2 = 1$: $T(t^2) = 3(0) + (5(0) - 2(0)) t + (4(0) + 1) t^2 = t^2$ So, $T(t^2) = t^2$, which is written as $0 \cdot 1 + 0 \cdot t + 1 \cdot t^2$.

3. Write the matrix representation:

   Now, the matrix representation of $T$ relative to the basis $\{ 1, t, t^2 \}$ is formed by expressing the result of each $T(1)$, $T(t)$, and $T(t^2)$ in terms of the basis elements:

   T(1) & T(t) & T(t^2) \end{bmatrix} = \begin{bmatrix} 3 & 0 & 0 \\ 5 & -2 & 0 \\ 0 & 4 & 1 \end{bmatrix}$$

Final Answer:

The matrix representation of $T$ relative to the basis $B = \{1, t, t^2\}$ is:

\begin{bmatrix} 3 & 0 & 0 \\ 5 & -2 & 0 \\ 0 & 4 & 1 \end{bmatrix} }$$ Do you want further details or have any questions? Here are 5 related questions: 1. How would you find the inverse of a transformation matrix? 2. What is the rank of the matrix representation of $$T$$? 3. How can you determine if the transformation $$T$$ is invertible? 4. What is the image and kernel of the transformation $$T$$? 5. How do eigenvalues and eigenvectors relate to the matrix representation of $$T$$? **Tip:** When finding matrix representations, always express the result in terms of the given basis.