Consider the matrix A = a 11 A12 A13 d14 a21 a22 a23 a24) show that the product L(X)=AX is a linear transformation, where šER. a22 a23 d24/
Consider the matrix A = a 11 A12 A13 d14 a21 a22 a23 a24) show that the product L(X)=AX is a linear transformation, where šER. a22 a23 d24/
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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![**Matrix and Linear Transformation Explanation**
Consider the matrix \( A \):
\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24}
\end{pmatrix}
\]
We need to show that the product transformation \( \mathbf{L}(\vec{x}) = A \vec{x} \) is a linear transformation, where \( \vec{x} \in \mathbb{R}^4 \).
**Explanation:**
1. **Matrix Representation:**
- The matrix \( A \) is a \( 2 \times 4 \) matrix consisting of elements \( a_{ij} \), where \( i \) represents the row and \( j \) represents the column.
2. **Transformation Description:**
- The function \( \mathbf{L}(\vec{x}) \) represents the transformation of a vector \( \vec{x} \) in the space \( \mathbb{R}^4 \) by the matrix \( A \).
3. **Linear Transformation Criteria:**
- A transformation is linear if it satisfies both additivity and scalar multiplication:
- **Additivity:** \( \mathbf{L}(\vec{x} + \vec{y}) = \mathbf{L}(\vec{x}) + \mathbf{L}(\vec{y}) \)
- **Scalar Multiplication:** \( \mathbf{L}(c \vec{x}) = c \mathbf{L}(\vec{x}) \) for any scalar \( c \).
4. **Application:**
- Showing \( \mathbf{L}(\vec{x}) \) is linear involves verifying these properties using matrix multiplication with vector inputs.
By demonstrating these properties, you can establish that \( \mathbf{L}(\vec{x}) = A \vec{x} \) is indeed a linear transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F486b24c8-8b02-4a6d-85d6-6aef3a827ba3%2F6e50b90c-1382-4b26-90b6-3de7d5307581%2Fx8d67io_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix and Linear Transformation Explanation**
Consider the matrix \( A \):
\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24}
\end{pmatrix}
\]
We need to show that the product transformation \( \mathbf{L}(\vec{x}) = A \vec{x} \) is a linear transformation, where \( \vec{x} \in \mathbb{R}^4 \).
**Explanation:**
1. **Matrix Representation:**
- The matrix \( A \) is a \( 2 \times 4 \) matrix consisting of elements \( a_{ij} \), where \( i \) represents the row and \( j \) represents the column.
2. **Transformation Description:**
- The function \( \mathbf{L}(\vec{x}) \) represents the transformation of a vector \( \vec{x} \) in the space \( \mathbb{R}^4 \) by the matrix \( A \).
3. **Linear Transformation Criteria:**
- A transformation is linear if it satisfies both additivity and scalar multiplication:
- **Additivity:** \( \mathbf{L}(\vec{x} + \vec{y}) = \mathbf{L}(\vec{x}) + \mathbf{L}(\vec{y}) \)
- **Scalar Multiplication:** \( \mathbf{L}(c \vec{x}) = c \mathbf{L}(\vec{x}) \) for any scalar \( c \).
4. **Application:**
- Showing \( \mathbf{L}(\vec{x}) \) is linear involves verifying these properties using matrix multiplication with vector inputs.
By demonstrating these properties, you can establish that \( \mathbf{L}(\vec{x}) = A \vec{x} \) is indeed a linear transformation.
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