Give a counterexample, with two numerical pairs (x₁, y₁) and (x₂, y₂), to show that the given transformation is not a linear transformation. +[x]-[x] (X₁x Y₁r X₂x Y₂) = (
Give a counterexample, with two numerical pairs (x₁, y₁) and (x₂, y₂), to show that the given transformation is not a linear transformation. +[x]-[x] (X₁x Y₁r X₂x Y₂) = (
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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![**Exercise: Counterexample for Linear Transformation**
**Objective:** Demonstrate that the given transformation is not linear by providing a counterexample with two numerical pairs \((x_1, y_1)\) and \((x_2, y_2)\).
**Transformation Definition:**
\[ T \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} y \\ x^2 \end{array} \right] \]
**Counterexample:**
Choose numerical values for \( (x_1, y_1, x_2, y_2) \) and illustrate how they fit into the transformation. Use these values to verify that the transformation does not satisfy the conditions for linearity (additivity and scalar multiplication).
\[ (x_1, y_1, x_2, y_2) = \left( \begin{array}{c} \text{[Blank for student input]} \end{array} \right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c4c6df5-8b96-455e-b2de-04bdb043521a%2Fd47c09dc-a083-4d55-8afc-640abeea9f64%2F0nv8ppf_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise: Counterexample for Linear Transformation**
**Objective:** Demonstrate that the given transformation is not linear by providing a counterexample with two numerical pairs \((x_1, y_1)\) and \((x_2, y_2)\).
**Transformation Definition:**
\[ T \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} y \\ x^2 \end{array} \right] \]
**Counterexample:**
Choose numerical values for \( (x_1, y_1, x_2, y_2) \) and illustrate how they fit into the transformation. Use these values to verify that the transformation does not satisfy the conditions for linearity (additivity and scalar multiplication).
\[ (x_1, y_1, x_2, y_2) = \left( \begin{array}{c} \text{[Blank for student input]} \end{array} \right) \]
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