Find the matrix A' for T relative to the basis B'. T: R3 → R3, T(x, y, z) = (x – y + 8z, 8x + y – z, x + 8y + z), B' = {(1, 0, 1), (0, 2, 2), (1, 2, 0)}

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Problem Statement:**

Find the matrix \( A' \) for \( T \) relative to the basis \( B' \).

**Transformation:**

\( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3, \; T(x, y, z) = (x - y + 8z, \; 8x + y - z, \; x + 8y + z) \)

**Basis:**

\( B' = \{(1, 0, 1), \; (0, 2, 2), \; (1, 2, 0)\} \)

**Matrix Representation:**

\[ A' = \begin{bmatrix} 
\text{\textvisiblespace} & \text{\textvisiblespace} & \text{\textvisiblespace} \\ 
\text{\textvisiblespace} & \text{\textvisiblespace} & \text{\textvisiblespace} \\ 
\text{\textvisiblespace} & \text{\textvisiblespace} & \text{\textvisiblespace} \end{bmatrix} \]

- Arrows and brackets suggest that you need to find the elements of matrix \( A' \) using the given linear transformation \( T \) and the basis \( B' \).
Transcribed Image Text:**Problem Statement:** Find the matrix \( A' \) for \( T \) relative to the basis \( B' \). **Transformation:** \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3, \; T(x, y, z) = (x - y + 8z, \; 8x + y - z, \; x + 8y + z) \) **Basis:** \( B' = \{(1, 0, 1), \; (0, 2, 2), \; (1, 2, 0)\} \) **Matrix Representation:** \[ A' = \begin{bmatrix} \text{\textvisiblespace} & \text{\textvisiblespace} & \text{\textvisiblespace} \\ \text{\textvisiblespace} & \text{\textvisiblespace} & \text{\textvisiblespace} \\ \text{\textvisiblespace} & \text{\textvisiblespace} & \text{\textvisiblespace} \end{bmatrix} \] - Arrows and brackets suggest that you need to find the elements of matrix \( A' \) using the given linear transformation \( T \) and the basis \( B' \).
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