Assume the mapping T W P2 ! P2 defined by
T .a0 C a1t C a2t
2
/ D 3a0 C .5a0 ￾ 2a1/t C .4a1 C a2/t 2
is linear. Find the matrix representation of T relative to the
basis B D f1; t; t 2
g

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:

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

### Final Answer:
The matrix representation of \( T \) relative to the basis \( B = \{1, t, t^2\} \) is:
\[
\boxed{
\begin{bmatrix}
3 & 0 & 0 \\
5 & -2 & 0 \\
0 & 4 & 1
\end{bmatrix}
}
\]

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.

Assume the mapping T W P2 ! P2 defined by T .a0 C a1t C a2t^2 / D 3a0 C .5a0 − 2a1/t C .4a1 C a2/t^2 is linear. Find the matrix representation of T relative to the basis B = {1, t, t^2}

Find the matrix representation of the linear transformation T relative to the polynomial basis B = {1, t, t^2}. This detailed solution covers linear transformations, polynomial bases, and matrix representations, suitable for undergraduate students studying linear algebra.

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