Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement:**
Let \(\mathbf{u} = (u_1, u_2) \in \mathbb{R}^2\), show that the transformation \(\mathbf{T}(u_1, u_2) = (u_1 + u_2, u_1 - 3u_2)\) is a linear transformation.
**Explanation:**
To prove that \(\mathbf{T}(u_1, u_2)\) is a linear transformation, we need to verify two main properties of linear transformations:
1. **Additivity:** \(\mathbf{T}(\mathbf{u} + \mathbf{v}) = \mathbf{T}(\mathbf{u}) + \mathbf{T}(\mathbf{v})\)
For \(\mathbf{u} = (u_1, u_2)\) and \(\mathbf{v} = (v_1, v_2)\),
\[
\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2)
\]
Applying the transformation,
\[
\mathbf{T}(\mathbf{u} + \mathbf{v}) = ((u_1 + v_1) + (u_2 + v_2), (u_1 + v_1) - 3(u_2 + v_2))
\]
Simplifying,
\[
= (u_1 + u_2 + v_1 + v_2, u_1 - 3u_2 + v_1 - 3v_2)
\]
Now check \(\mathbf{T}(\mathbf{u}) + \mathbf{T}(\mathbf{v})\),
\[
\mathbf{T}(\mathbf{u}) = (u_1 + u_2, u_1 - 3u_2)
\]
\[
\mathbf{T}(\mathbf{v}) = (v_1 + v_2, v_1 - 3v_2)
\]
Adding these,
\[
\mathbf{T}(\mathbf{u}) + \mathbf{T}(\mathbf{v}) = (u_1 + u_2 + v_1 + v](https://content.bartleby.com/qna-images/question/80a13d8c-5188-4010-ba23-710cc489cc25/7565f9d4-a5cc-48e5-8e13-fd22c1439927/lob9stu_processed.png)
Transcribed Image Text:**Problem Statement:**
Let \(\mathbf{u} = (u_1, u_2) \in \mathbb{R}^2\), show that the transformation \(\mathbf{T}(u_1, u_2) = (u_1 + u_2, u_1 - 3u_2)\) is a linear transformation.
**Explanation:**
To prove that \(\mathbf{T}(u_1, u_2)\) is a linear transformation, we need to verify two main properties of linear transformations:
1. **Additivity:** \(\mathbf{T}(\mathbf{u} + \mathbf{v}) = \mathbf{T}(\mathbf{u}) + \mathbf{T}(\mathbf{v})\)
For \(\mathbf{u} = (u_1, u_2)\) and \(\mathbf{v} = (v_1, v_2)\),
\[
\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2)
\]
Applying the transformation,
\[
\mathbf{T}(\mathbf{u} + \mathbf{v}) = ((u_1 + v_1) + (u_2 + v_2), (u_1 + v_1) - 3(u_2 + v_2))
\]
Simplifying,
\[
= (u_1 + u_2 + v_1 + v_2, u_1 - 3u_2 + v_1 - 3v_2)
\]
Now check \(\mathbf{T}(\mathbf{u}) + \mathbf{T}(\mathbf{v})\),
\[
\mathbf{T}(\mathbf{u}) = (u_1 + u_2, u_1 - 3u_2)
\]
\[
\mathbf{T}(\mathbf{v}) = (v_1 + v_2, v_1 - 3v_2)
\]
Adding these,
\[
\mathbf{T}(\mathbf{u}) + \mathbf{T}(\mathbf{v}) = (u_1 + u_2 + v_1 + v
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