**Linear Transformation from $ \mathbb{R}^3 $ to $ \mathbb{R}^2 $**Define the transformation $ T: \mathbb{R}^3 \to \mathbb{R}^2 $ by:\[T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 - 6x_2 \\ 2x_1 - 12x_2 \end{bmatrix}\]**Range of T**The range of $ T $ is determined by the span of vectors resulting from the transformation. Choose the correct option that defines the span:- $ \begin{bmatrix} \text{[Enter the components]} \end{bmatrix} $- $\{0\}$- $\mathbb{R}^2$To conclude, the span of a single vector is highlighted as the range of $ T $.

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Answered: Define T:R –→ R² by T ¤1 – 6x2 12 2x1 –…

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Define T:R –→ R² by T ¤1 – 6x2 12 2x1 – 12x2 . The range of T is The span of a single vector: O {0} OR?

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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**Linear Transformation from $ \mathbb{R}^3 $ to $ \mathbb{R}^2 $**

Define the transformation $ T: \mathbb{R}^3 \to \mathbb{R}^2 $ by:

\[
T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 - 6x_2 \\ 2x_1 - 12x_2 \end{bmatrix}
\]

**Range of T**

The range of $ T $ is determined by the span of vectors resulting from the transformation. Choose the correct option that defines the span:

- $ \begin{bmatrix} \text{[Enter the components]} \end{bmatrix} $
- $\{0\}$
- $\mathbb{R}^2$

To conclude, the span of a single vector is highlighted as the range of $ T $.

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