Let T: P₂ → P₁ be defined by T(p(x)) = p′ (x). Let B = { 1 + x + x², 1 + x, 1} and C = {1 + 3x, 2 + 5x}. Find [T], the matrix representation of T with respect to the bases B and C. 177² - [Ex 5 ]]
Let T: P₂ → P₁ be defined by T(p(x)) = p′ (x). Let B = { 1 + x + x², 1 + x, 1} and C = {1 + 3x, 2 + 5x}. Find [T], the matrix representation of T with respect to the bases B and C. 177² - [Ex 5 ]]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hi can I please get some assiatance with this problem, I'm a bit stuck on the steps. Thank you.
![### Linear Transformations and Matrix Representations
**Problem Statement:**
Let \( T : P_2 \rightarrow P_1 \) be defined by \( T(p(x)) = p'(x) \).
Let \( \mathcal{B} = \{1 + x + x^2, 1 + x, 1\} \) and \( \mathcal{C} = \{1 + 3x, 2 + 5x\} \).
Find \([T]_{\mathcal{B}}^{\mathcal{C}}\), the matrix representation of \(T\) with respect to the bases \(\mathcal{B}\) and \(\mathcal{C}\).
\[
[T]_{\mathcal{B}}^{\mathcal{C}} =
\begin{bmatrix}
\text{Ex: 5} & & \\
& & \\
\end{bmatrix}
\]
**Explanation:**
- **Transformation Definition:**
\( T \) is a linear transformation that maps a second-degree polynomial \( p(x) \) to its first derivative \( p'(x) \).
- **Basis \(\mathcal{B}\):**
The basis \(\mathcal{B}\) for \( P_2 \) (polynomials of degree at most 2) is given as the set \(\{1 + x + x^2, 1 + x, 1\} \).
- **Basis \(\mathcal{C}\):**
The basis \(\mathcal{C}\) for \( P_1 \) (polynomials of degree at most 1) is given as the set \(\{1 + 3x, 2 + 5x\} \).
- **Goal:**
To find the matrix representation \([T]_{\mathcal{B}}^{\mathcal{C}}\) in the specified bases.
**Steps to Solve:**
1. **Compute the Derivatives:**
Apply \( T \) to each element in \(\mathcal{B}\):
- \( T(1 + x + x^2) = (1 + x + x^2)' = 1 + 2x \)
- \( T(1 + x) = (1 + x)' = 1 \)
- \( T(1) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8aa82a-31a1-4382-a30e-3ad6a3174f84%2Fc5116619-46c8-4932-ae07-6295788507a9%2Fvmhrrvs_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformations and Matrix Representations
**Problem Statement:**
Let \( T : P_2 \rightarrow P_1 \) be defined by \( T(p(x)) = p'(x) \).
Let \( \mathcal{B} = \{1 + x + x^2, 1 + x, 1\} \) and \( \mathcal{C} = \{1 + 3x, 2 + 5x\} \).
Find \([T]_{\mathcal{B}}^{\mathcal{C}}\), the matrix representation of \(T\) with respect to the bases \(\mathcal{B}\) and \(\mathcal{C}\).
\[
[T]_{\mathcal{B}}^{\mathcal{C}} =
\begin{bmatrix}
\text{Ex: 5} & & \\
& & \\
\end{bmatrix}
\]
**Explanation:**
- **Transformation Definition:**
\( T \) is a linear transformation that maps a second-degree polynomial \( p(x) \) to its first derivative \( p'(x) \).
- **Basis \(\mathcal{B}\):**
The basis \(\mathcal{B}\) for \( P_2 \) (polynomials of degree at most 2) is given as the set \(\{1 + x + x^2, 1 + x, 1\} \).
- **Basis \(\mathcal{C}\):**
The basis \(\mathcal{C}\) for \( P_1 \) (polynomials of degree at most 1) is given as the set \(\{1 + 3x, 2 + 5x\} \).
- **Goal:**
To find the matrix representation \([T]_{\mathcal{B}}^{\mathcal{C}}\) in the specified bases.
**Steps to Solve:**
1. **Compute the Derivatives:**
Apply \( T \) to each element in \(\mathcal{B}\):
- \( T(1 + x + x^2) = (1 + x + x^2)' = 1 + 2x \)
- \( T(1 + x) = (1 + x)' = 1 \)
- \( T(1) =
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