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 Transformations**
A linear transformation is a mathematical function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. The transformation T: ℝ² → ℝ² defines a mapping from a two-dimensional vector space to another two-dimensional vector space.
**Problem Statement:**
Let T: ℝ² → ℝ² be a linear transformation such that T(x₁, x₂) = (x₁ + x₂, 6x₁ + 2x₂). Find x such that T(x) = (1, 22).
**Solution Approach:**
1. You start with the given transformation:
\[
T(x_1, x_2) = (x_1 + x_2, 6x_1 + 2x_2)
\]
2. We need to find a vector \( x = (x_1, x_2) \) such that
\[
T(x) = (1, 22)
\]
3. This gives us two equations to solve:
- \( x_1 + x_2 = 1 \)
- \( 6x_1 + 2x_2 = 22 \)
4. Solve these equations simultaneously to find \( x_1 \) and \( x_2 \).
Therefore, you need to solve the system of linear equations to find the vector \( x \) in ℝ² that satisfies the condition given by the transformation T.
**Graphical or Diagrammatic Explanation:**
- There is no diagram or graph provided in this problem.
- This problem focuses on solving the set of linear equations derived from the given transformation.
**Input Box:**
- There is an input box represented by \( x = \boxed{\phantom{x}} \) where you will enter the solution vector \( x \).
Make sure to follow the conditions set by the linear transformation and solve the system of linear equations accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2F2cce465d-e2ef-4951-a69c-d80bd99d962f%2Fm5v7sjo_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Transformations**
A linear transformation is a mathematical function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. The transformation T: ℝ² → ℝ² defines a mapping from a two-dimensional vector space to another two-dimensional vector space.
**Problem Statement:**
Let T: ℝ² → ℝ² be a linear transformation such that T(x₁, x₂) = (x₁ + x₂, 6x₁ + 2x₂). Find x such that T(x) = (1, 22).
**Solution Approach:**
1. You start with the given transformation:
\[
T(x_1, x_2) = (x_1 + x_2, 6x_1 + 2x_2)
\]
2. We need to find a vector \( x = (x_1, x_2) \) such that
\[
T(x) = (1, 22)
\]
3. This gives us two equations to solve:
- \( x_1 + x_2 = 1 \)
- \( 6x_1 + 2x_2 = 22 \)
4. Solve these equations simultaneously to find \( x_1 \) and \( x_2 \).
Therefore, you need to solve the system of linear equations to find the vector \( x \) in ℝ² that satisfies the condition given by the transformation T.
**Graphical or Diagrammatic Explanation:**
- There is no diagram or graph provided in this problem.
- This problem focuses on solving the set of linear equations derived from the given transformation.
**Input Box:**
- There is an input box represented by \( x = \boxed{\phantom{x}} \) where you will enter the solution vector \( x \).
Make sure to follow the conditions set by the linear transformation and solve the system of linear equations accurately.
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