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Let T: R² →R² be a linear transformation such that T (×₁,×₂) = (x₁ + x₂, 2x₁ +6x₂). Find x such that T(x)=(5,6). x = (9,-4)

Let T: R² →R² be a linear transformation such that T (×₁,×₂) = (x₁ + x₂, 2x₁ +6x₂). Find x such that T(x)=(5,6). x = (9,-4)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Linear Transformation in \(\mathbb{R}^2\) #### Problem Statement Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation such that \[ T(x_1, x_2) = (x_1 + x_2, 2x_1 + 6x_2). \] Find \( \mathbf{x} \) such that \[ T(\mathbf{x}) = (5, -6). \] *Solution:* \[ \mathbf{x} = (9, -4). \] --- ### Explanation: To find the vector \(\mathbf{x} = (x_1, x_2)\) such that \(T(\mathbf{x}) = (5, -6)\), we start by writing down the transformation equations: \[ T(x_1, x_2) = (x_1 + x_2, 2x_1 + 6x_2). \] We need to find \(x_1\) and \(x_2\) such that: \[ x_1 + x_2 = 5 \] \[ 2x_1 + 6x_2 = -6. \] Solve these equations simultaneously: 1. \(x_1 + x_2 = 5\) 2. \(2x_1 + 6x_2 = -6\) From the first equation: \[ x_1 = 5 - x_2. \] Substitute \(x_1\) in the second equation: \[ 2(5 - x_2) + 6x_2 = -6. \] Simplify: \[ 10 - 2x_2 + 6x_2 = -6, \] \[ 10 + 4x_2 = -6, \] \[ 4x_2 = -6 - 10, \] \[ 4x_2 = -16, \] \[ x_2 = -4. \] Substitute \(x_2 = -4\) back into the first equation: \[ x_1 = 5 - (-4), \] \[ x_1 = 9. \] So, \(\mathbf{x} = (9, -4)\). ### Conclusion Thus, the vector \
Transcribed Image Text:### Linear Transformation in \(\mathbb{R}^2\) #### Problem Statement Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation such that \[ T(x_1, x_2) = (x_1 + x_2, 2x_1 + 6x_2). \] Find \( \mathbf{x} \) such that \[ T(\mathbf{x}) = (5, -6). \] *Solution:* \[ \mathbf{x} = (9, -4). \] --- ### Explanation: To find the vector \(\mathbf{x} = (x_1, x_2)\) such that \(T(\mathbf{x}) = (5, -6)\), we start by writing down the transformation equations: \[ T(x_1, x_2) = (x_1 + x_2, 2x_1 + 6x_2). \] We need to find \(x_1\) and \(x_2\) such that: \[ x_1 + x_2 = 5 \] \[ 2x_1 + 6x_2 = -6. \] Solve these equations simultaneously: 1. \(x_1 + x_2 = 5\) 2. \(2x_1 + 6x_2 = -6\) From the first equation: \[ x_1 = 5 - x_2. \] Substitute \(x_1\) in the second equation: \[ 2(5 - x_2) + 6x_2 = -6. \] Simplify: \[ 10 - 2x_2 + 6x_2 = -6, \] \[ 10 + 4x_2 = -6, \] \[ 4x_2 = -6 - 10, \] \[ 4x_2 = -16, \] \[ x_2 = -4. \] Substitute \(x_2 = -4\) back into the first equation: \[ x_1 = 5 - (-4), \] \[ x_1 = 9. \] So, \(\mathbf{x} = (9, -4)\). ### Conclusion Thus, the vector \
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