p(1) p (1) [p(2) p'(2)] 1. Let T: P₂ →M²×² be defined by T (p(x))= relative to the following bases. (i.e. find [7]}') B=(x+1, 2x²+x+2, x²+2x+2) and B' = Find the matrix of T 2 2 (16 36 36 3
p(1) p (1) [p(2) p'(2)] 1. Let T: P₂ →M²×² be defined by T (p(x))= relative to the following bases. (i.e. find [7]}') B=(x+1, 2x²+x+2, x²+2x+2) and B' = Find the matrix of T 2 2 (16 36 36 3
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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Please provide detailed explantion with english as well.
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![## Linear Algebra: Transformations and Matrices
### Problem Statement
1. Let \( T: P_2 \to M^{2 \times 2} \) be defined by:
\[ T(p(x)) = \begin{bmatrix}
p(1) & p'(1) \\
p(2) & p'(2)
\end{bmatrix} \]
Find the matrix of \( T \) relative to the following bases (i.e., find \( [T]^B_{B'} \)).
Given Bases:
\[ B = \{x + 1, 2x^2 + x + 2, x^2 + 2x + 2\} \]
\[ B' = \left\{
\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix},
\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix},
\begin{bmatrix} 1 & 2 \\ 2 & -2 \end{bmatrix},
\begin{bmatrix} 2 & 2 \\ 1 & 1 \end{bmatrix}
\right\} \]
### Explanation
- **Transformation \( T \)**: This transformation maps polynomials of degree at most 2 (i.e., polynomials in \( P_2 \)) to 2x2 matrices.
- **Matrix Representation**: The goal is to express the linear transformation \( T \) in matrix form. This involves finding a matrix representation of \( T \) relative to given bases \( B \) and \( B' \).
- **Bases \( B \) and \( B' \)**:
- \( B \) is a basis for the polynomial space \( P_2 \) and consists of three polynomials.
- \( B' \) is a basis for the space of \( 2 \times 2 \) matrices and consists of four matrices.
### Steps to Solve:
1. **Apply \( T \) to Each Basis Element of \( B \)**:
- Find the image of each basis polynomial under \( T \).
2. **Express Resulting Matrices in Terms of \( B' \)**:
- Write each result as a linear combination of the matrices in \( B' \).
3. **Form the Matrix Representation**:
- Use](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe354d500-dc10-4e3d-a060-42009e500b8b%2F3c9bf6bf-af29-4a32-ac89-7a9fe2539ca8%2Fj6pb1pg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Linear Algebra: Transformations and Matrices
### Problem Statement
1. Let \( T: P_2 \to M^{2 \times 2} \) be defined by:
\[ T(p(x)) = \begin{bmatrix}
p(1) & p'(1) \\
p(2) & p'(2)
\end{bmatrix} \]
Find the matrix of \( T \) relative to the following bases (i.e., find \( [T]^B_{B'} \)).
Given Bases:
\[ B = \{x + 1, 2x^2 + x + 2, x^2 + 2x + 2\} \]
\[ B' = \left\{
\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix},
\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix},
\begin{bmatrix} 1 & 2 \\ 2 & -2 \end{bmatrix},
\begin{bmatrix} 2 & 2 \\ 1 & 1 \end{bmatrix}
\right\} \]
### Explanation
- **Transformation \( T \)**: This transformation maps polynomials of degree at most 2 (i.e., polynomials in \( P_2 \)) to 2x2 matrices.
- **Matrix Representation**: The goal is to express the linear transformation \( T \) in matrix form. This involves finding a matrix representation of \( T \) relative to given bases \( B \) and \( B' \).
- **Bases \( B \) and \( B' \)**:
- \( B \) is a basis for the polynomial space \( P_2 \) and consists of three polynomials.
- \( B' \) is a basis for the space of \( 2 \times 2 \) matrices and consists of four matrices.
### Steps to Solve:
1. **Apply \( T \) to Each Basis Element of \( B \)**:
- Find the image of each basis polynomial under \( T \).
2. **Express Resulting Matrices in Terms of \( B' \)**:
- Write each result as a linear combination of the matrices in \( B' \).
3. **Form the Matrix Representation**:
- Use
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