Draw the image of the unit square (shown in green) under the transformation matrix A =
Draw the image of the unit square (shown in green) under the transformation matrix A =
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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![## Transformation of the Unit Square
**Objective:**
Draw the image of the unit square, shown in green on the grid, under the transformation matrix \( A = \begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \).
**Instructions:**
1. **Identify the Unit Square:**
- The unit square is positioned with its bottom-left corner at the origin (0,0) and occupies the coordinates (0,0), (1,0), (1,1), and (0,1).
2. **Apply the Transformation Matrix:**
- Use the matrix \( A = \begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \) to transform each vertex of the unit square.
3. **Calculate New Coordinates:**
- Calculate the new coordinates for each vertex by multiplying each vertex coordinate by matrix \( A \).
For the vertices:
- (0,0):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}
\]
- (1,0):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \end{bmatrix}
\]
- (1,1):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}
\]
- (0,1):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5025798a-663c-4f82-a82a-d0677c58328a%2Fd186f144-cf55-4327-aace-a52197819b92%2Ffzeewss_processed.png&w=3840&q=75)
Transcribed Image Text:## Transformation of the Unit Square
**Objective:**
Draw the image of the unit square, shown in green on the grid, under the transformation matrix \( A = \begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \).
**Instructions:**
1. **Identify the Unit Square:**
- The unit square is positioned with its bottom-left corner at the origin (0,0) and occupies the coordinates (0,0), (1,0), (1,1), and (0,1).
2. **Apply the Transformation Matrix:**
- Use the matrix \( A = \begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \) to transform each vertex of the unit square.
3. **Calculate New Coordinates:**
- Calculate the new coordinates for each vertex by multiplying each vertex coordinate by matrix \( A \).
For the vertices:
- (0,0):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}
\]
- (1,0):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \end{bmatrix}
\]
- (1,1):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}
\]
- (0,1):
\[
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}
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