Let e₁ = 6 [1], e₂ = [9], Y₁ = [3], and y₂ = [¹], and let T.: R² → R²be a linear transformation that maps e₁ into y₁ and maps e2 into y₂. Find the image of [53] ² and [₁].
Let e₁ = 6 [1], e₂ = [9], Y₁ = [3], and y₂ = [¹], and let T.: R² → R²be a linear transformation that maps e₁ into y₁ and maps e2 into y₂. Find the image of [53] ² and [₁].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Linear Transformation
![**Problem Statement:**
Given the vectors:
- \( \mathbf{e_1} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)
- \( \mathbf{e_2} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)
- \( \mathbf{y_1} = \begin{bmatrix} 2 \\ 5 \end{bmatrix} \)
- \( \mathbf{y_2} = \begin{bmatrix} -1 \\ 6 \end{bmatrix} \)
and let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation that maps \( \mathbf{e_1} \) into \( \mathbf{y_1} \) and maps \( \mathbf{e_2} \) into \( \mathbf{y_2} \). Find the image of
\[
\begin{bmatrix} 5 \\ -3 \end{bmatrix}
\]
and
\[
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}.
\]
**Explanation:**
1. **Determining the Linear Transformation Matrix**:
- The linear transformation \( T \) is defined by the basis vectors \( \mathbf{e_1} \) and \( \mathbf{e_2} \), and their corresponding images \( \mathbf{y_1} \) and \( \mathbf{y_2} \).
- The matrix representation of \( T \) in the standard basis can be found by using the columns \( \mathbf{y_1} \) and \( \mathbf{y_2} \):
\[ T = \begin{bmatrix} \mathbf{y_1} & \mathbf{y_2} \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 5 & 6 \end{bmatrix} \]
2. **Finding the Image of \( \mathbf{v} = \begin{bmatrix} 5 \\ -3 \end{bmatrix} \)**:
- To find the image of \( \mathbf{v} \) under \( T \), we multiply the transformation matrix by \( \mathbf{v} \):
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9baea4c0-9d9e-491a-8271-bc2a2bbd8b45%2Fe0826218-5f9a-469b-b29e-e0ee895e9abc%2Fow6hfzah_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the vectors:
- \( \mathbf{e_1} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)
- \( \mathbf{e_2} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)
- \( \mathbf{y_1} = \begin{bmatrix} 2 \\ 5 \end{bmatrix} \)
- \( \mathbf{y_2} = \begin{bmatrix} -1 \\ 6 \end{bmatrix} \)
and let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation that maps \( \mathbf{e_1} \) into \( \mathbf{y_1} \) and maps \( \mathbf{e_2} \) into \( \mathbf{y_2} \). Find the image of
\[
\begin{bmatrix} 5 \\ -3 \end{bmatrix}
\]
and
\[
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}.
\]
**Explanation:**
1. **Determining the Linear Transformation Matrix**:
- The linear transformation \( T \) is defined by the basis vectors \( \mathbf{e_1} \) and \( \mathbf{e_2} \), and their corresponding images \( \mathbf{y_1} \) and \( \mathbf{y_2} \).
- The matrix representation of \( T \) in the standard basis can be found by using the columns \( \mathbf{y_1} \) and \( \mathbf{y_2} \):
\[ T = \begin{bmatrix} \mathbf{y_1} & \mathbf{y_2} \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 5 & 6 \end{bmatrix} \]
2. **Finding the Image of \( \mathbf{v} = \begin{bmatrix} 5 \\ -3 \end{bmatrix} \)**:
- To find the image of \( \mathbf{v} \) under \( T \), we multiply the transformation matrix by \( \mathbf{v} \):
\[
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