Let ü=(u,u)eR², show that T(u1,u2)=(u1+ U2, U̟– 3u2) is a linear transformation.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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
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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