Tis a linear transformation from R² into R² as shown below. The standard matrix for T-l is T'x, у) %3D ( Зх + 4y, 5x+7у)
Tis a linear transformation from R² into R² as shown below. The standard matrix for T-l is T'x, у) %3D ( Зх + 4y, 5x+7у)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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T is a linear transformation from R2 into R2 as shown below. (see image) The standard matrix for T-1 is:
![### Linear Transformation in \( \mathbb{R}^2 \)
#### Transformation Description
T is a linear transformation from \( \mathbb{R}^2 \) into \( \mathbb{R}^2 \) as described below.
The standard matrix for \( T^{-1} \) is:
\[
T(x, y) = (3x + 4y, \; 5x + 7y)
\]
In this linear transformation, a vector \( (x, y) \) in \( \mathbb{R}^2 \) is mapped to a new vector \( (3x + 4y, \; 5x + 7y) \).
### Detailed Explanation
- **Domain and Co-domain:**
- The domain of the transformation T is \( \mathbb{R}^2 \), which means it takes input vectors from a 2-dimensional space.
- The co-domain where the transformation maps the vectors is also \( \mathbb{R}^2 \).
- **Transformation Function:**
- The transformation function \( T \) described by the equations \( 3x + 4y \) and \( 5x + 7y \) suggests a linear combination of the input vector components \( (x, y) \).
- A vector \( (x, y) \) under transformation \( T \) results in another vector \( (3x + 4y, \; 5x + 7y) \).
- The coefficients \( 3, 4, 5 \), and \( 7 \) represent how much each component of the input vector contributes to the components of the output vector.
Understanding this transformation is foundational in linear algebra as it demonstrates how vectors in two-dimensional space can be linearly mapped to other vectors in the same space using a transformation matrix. This application has broad implications in fields such as computer graphics, engineering, and physics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc17e2a4-76f7-46f2-97e3-7d367a16a370%2F4edbebb0-911e-486f-89ec-667a071d5e38%2Fd5aihk.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation in \( \mathbb{R}^2 \)
#### Transformation Description
T is a linear transformation from \( \mathbb{R}^2 \) into \( \mathbb{R}^2 \) as described below.
The standard matrix for \( T^{-1} \) is:
\[
T(x, y) = (3x + 4y, \; 5x + 7y)
\]
In this linear transformation, a vector \( (x, y) \) in \( \mathbb{R}^2 \) is mapped to a new vector \( (3x + 4y, \; 5x + 7y) \).
### Detailed Explanation
- **Domain and Co-domain:**
- The domain of the transformation T is \( \mathbb{R}^2 \), which means it takes input vectors from a 2-dimensional space.
- The co-domain where the transformation maps the vectors is also \( \mathbb{R}^2 \).
- **Transformation Function:**
- The transformation function \( T \) described by the equations \( 3x + 4y \) and \( 5x + 7y \) suggests a linear combination of the input vector components \( (x, y) \).
- A vector \( (x, y) \) under transformation \( T \) results in another vector \( (3x + 4y, \; 5x + 7y) \).
- The coefficients \( 3, 4, 5 \), and \( 7 \) represent how much each component of the input vector contributes to the components of the output vector.
Understanding this transformation is foundational in linear algebra as it demonstrates how vectors in two-dimensional space can be linearly mapped to other vectors in the same space using a transformation matrix. This application has broad implications in fields such as computer graphics, engineering, and physics.
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