**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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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

Problem 1RQ

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Question

**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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