Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![The problem statement is as follows:
**Question 3:**
If reflection about the origin is represented by the matrix
\[ R_0 = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \]
reflect the letter whose coordinates are represented by the matrix
\[ A = \begin{bmatrix} 0 & 4 & 4 & 1 & 0 \\ 0 & 0 & 1 & 1 & 6 \end{bmatrix} \]
### Explanation:
This exercise involves matrix transformations to reflect a figure about the origin in a coordinate plane. The matrix \( R_0 \) is a reflection matrix which, when multiplied by a set of coordinates, reflects those coordinates about the origin.
- Matrix \( R_0 \) is defined as:
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \]
This matrix will change the sign of each coordinate, effectively reflecting them about the origin.
- Matrix \( A \) consists of coordinates of specific points forming a letter:
\[ \begin{bmatrix} 0 & 4 & 4 & 1 & 0 \\ 0 & 0 & 1 & 1 & 6 \end{bmatrix} \]
The columns represent points' coordinates \((x, y)\): (0,0), (4,0), (4,1), (1,1), and (0,6).
To find the reflected coordinates, multiply matrix \( R_0 \) by matrix \( A \):
\[ R_0 \times A = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 0 & 4 & 4 & 1 & 0 \\ 0 & 0 & 1 & 1 & 6 \end{bmatrix} \]
This will yield a new set of coordinates representing the reflected figure.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5b6e13e-b165-4b79-92da-bc958eab9f6d%2F3d28954b-cbd4-4b07-b179-169b33bcaa92%2Fvsjcmyf_processed.png&w=3840&q=75)
Transcribed Image Text:The problem statement is as follows:
**Question 3:**
If reflection about the origin is represented by the matrix
\[ R_0 = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \]
reflect the letter whose coordinates are represented by the matrix
\[ A = \begin{bmatrix} 0 & 4 & 4 & 1 & 0 \\ 0 & 0 & 1 & 1 & 6 \end{bmatrix} \]
### Explanation:
This exercise involves matrix transformations to reflect a figure about the origin in a coordinate plane. The matrix \( R_0 \) is a reflection matrix which, when multiplied by a set of coordinates, reflects those coordinates about the origin.
- Matrix \( R_0 \) is defined as:
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \]
This matrix will change the sign of each coordinate, effectively reflecting them about the origin.
- Matrix \( A \) consists of coordinates of specific points forming a letter:
\[ \begin{bmatrix} 0 & 4 & 4 & 1 & 0 \\ 0 & 0 & 1 & 1 & 6 \end{bmatrix} \]
The columns represent points' coordinates \((x, y)\): (0,0), (4,0), (4,1), (1,1), and (0,6).
To find the reflected coordinates, multiply matrix \( R_0 \) by matrix \( A \):
\[ R_0 \times A = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 0 & 4 & 4 & 1 & 0 \\ 0 & 0 & 1 & 1 & 6 \end{bmatrix} \]
This will yield a new set of coordinates representing the reflected figure.
![The problem presented is to find the variables in the matrix equation:
\[ \begin{bmatrix} 3x + y & -17 \\ 19 & 2 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & -17 \\ 19 & 2x + 3y \end{bmatrix} \]
This equation involves two equal 2x2 matrices. To find the variables \(x\) and \(y\), equate corresponding elements from each matrix:
1. \(3x + y = \frac{1}{2}\)
2. \(-17 = -17\) (always true)
3. \(19 = 19\) (always true)
4. \(2 = 2x + 3y\)
Solve these equations to find values for \(x\) and \(y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5b6e13e-b165-4b79-92da-bc958eab9f6d%2F3d28954b-cbd4-4b07-b179-169b33bcaa92%2Fljg4dsy_processed.png&w=3840&q=75)
Transcribed Image Text:The problem presented is to find the variables in the matrix equation:
\[ \begin{bmatrix} 3x + y & -17 \\ 19 & 2 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & -17 \\ 19 & 2x + 3y \end{bmatrix} \]
This equation involves two equal 2x2 matrices. To find the variables \(x\) and \(y\), equate corresponding elements from each matrix:
1. \(3x + y = \frac{1}{2}\)
2. \(-17 = -17\) (always true)
3. \(19 = 19\) (always true)
4. \(2 = 2x + 3y\)
Solve these equations to find values for \(x\) and \(y\).
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