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3. Consider the linear transformations T: R2 R2 and S: R2 R2 defined by X2 ¹(E)-E) (C)-Ld 3x₁ Find T-¹ and S-1. Using your answers, write a formula for (TS)-¹. T S

3. Consider the linear transformations T: R2 R2 and S: R2 R2 defined by X2 ¹(E)-E) (C)-Ld 3x₁ Find T-¹ and S-1. Using your answers, write a formula for (TS)-¹. T S

Advanced Engineering Mathematics
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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3. Consider the linear transformations \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) and \( S : \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \[ T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} 2x_2 \\ 3x_1 \end{bmatrix} \quad S \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} x_2 \\ x_1 - 3x_2 \end{bmatrix} . \] Find \( T^{-1} \) and \( S^{-1} \). Using your answers, write a formula for \( (TS)^{-1} \).
Transcribed Image Text:3. Consider the linear transformations \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) and \( S : \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \[ T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} 2x_2 \\ 3x_1 \end{bmatrix} \quad S \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} x_2 \\ x_1 - 3x_2 \end{bmatrix} . \] Find \( T^{-1} \) and \( S^{-1} \). Using your answers, write a formula for \( (TS)^{-1} \).
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