Describe geometrically what I does to vector x in R². 0 T(x) = 1 0 90 Reflection through the axis Reflection through the origin O Reflection through the line 1 — —£2 O contraction by-1 Reflection through the line #1 = 02
Describe geometrically what I does to vector x in R². 0 T(x) = 1 0 90 Reflection through the axis Reflection through the origin O Reflection through the line 1 — —£2 O contraction by-1 Reflection through the line #1 = 02
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
![### Understanding Linear Transformations
In this problem, we are asked to describe geometrically what the transformation \( T \) does to a vector \( \mathbf{x} \) in \(\mathbb{R}^2\).
The transformation is given as:
\[ T(\mathbf{x}) = \begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix} \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \]
This matrix transformation can be expanded to:
\[ T(\mathbf{x}) = \begin{bmatrix}
-1 \cdot x_1 + 0 \cdot x_2 \\
0 \cdot x_1 + (-1) \cdot x_2
\end{bmatrix} = \begin{bmatrix}
-x_1 \\
-x_2
\end{bmatrix} \]
### Multiple Choice Options
- \( \circ \) Reflection through the \( x \)-axis
- \( \circ \) Reflection through the origin
- \( \circ \) Reflection through the line \( x_1 = -x_2 \)
- \( \circ \) Contraction by \(-1\)
- \( \circ \) Reflection through the line \( x_1 = x_2 \)
### Analysis
- **Reflection through the \( x \)-axis**: This would change \( (x_1, x_2) \) to \( (x_1, -x_2) \).
- **Reflection through the origin**: This would change \( (x_1, x_2) \) to \( (-x_1, -x_2) \).
- **Reflection through the line \( x_1 = -x_2 \)**: This would swap the coordinates and negate one, giving \( (-x_2, -x_1) \).
- **Contraction by \(-1\)**: This seems ambiguous in this context and is not a standard geometric transformation term.
- **Reflection through the line \( x_1 = x_2 \)**: This would swap the coordinates \( (x_2, x_1) \) without negating them.
### Conclusion
Given the options, the correct geometric interpretation is:
\[ \boxed{\text{Reflection through the origin}} \]
Such a transformation results in each](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F157e66f5-e794-4af2-988e-1885b818a80a%2F1b91ddf0-125c-4692-a061-36fe01f5d4da%2Fko4zito_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Linear Transformations
In this problem, we are asked to describe geometrically what the transformation \( T \) does to a vector \( \mathbf{x} \) in \(\mathbb{R}^2\).
The transformation is given as:
\[ T(\mathbf{x}) = \begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix} \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \]
This matrix transformation can be expanded to:
\[ T(\mathbf{x}) = \begin{bmatrix}
-1 \cdot x_1 + 0 \cdot x_2 \\
0 \cdot x_1 + (-1) \cdot x_2
\end{bmatrix} = \begin{bmatrix}
-x_1 \\
-x_2
\end{bmatrix} \]
### Multiple Choice Options
- \( \circ \) Reflection through the \( x \)-axis
- \( \circ \) Reflection through the origin
- \( \circ \) Reflection through the line \( x_1 = -x_2 \)
- \( \circ \) Contraction by \(-1\)
- \( \circ \) Reflection through the line \( x_1 = x_2 \)
### Analysis
- **Reflection through the \( x \)-axis**: This would change \( (x_1, x_2) \) to \( (x_1, -x_2) \).
- **Reflection through the origin**: This would change \( (x_1, x_2) \) to \( (-x_1, -x_2) \).
- **Reflection through the line \( x_1 = -x_2 \)**: This would swap the coordinates and negate one, giving \( (-x_2, -x_1) \).
- **Contraction by \(-1\)**: This seems ambiguous in this context and is not a standard geometric transformation term.
- **Reflection through the line \( x_1 = x_2 \)**: This would swap the coordinates \( (x_2, x_1) \) without negating them.
### Conclusion
Given the options, the correct geometric interpretation is:
\[ \boxed{\text{Reflection through the origin}} \]
Such a transformation results in each
Transcribed Image Text:**Linear Algebra Solution Analysis**
**Problem Statement:**
Let \( A \) be a \( 2 \times 4 \) matrix with two pivot positions. Determine if:
a) The equation \( A\mathbf{x} = \mathbf{b} \) has a unique solution for every possible \(\mathbf{b}\).
**Options to Choose From:**
1. ⭕ The equation \( A\mathbf{x} = \mathbf{0} \) has a nontrivial solution and equation \( A\mathbf{x} = \mathbf{b} \) has a unique solution for every possible \(\mathbf{b}\).
2. ⭕ The equation \( A\mathbf{x} = \mathbf{0} \) has a nontrivial solution and equation \( A\mathbf{x} = \mathbf{b} \) does not have a unique solution for every possible \(\mathbf{b}\).
3. ⭕ The equation \( A\mathbf{x} = \mathbf{0} \) has a trivial solution and equation \( A\mathbf{x} = \mathbf{b} \) has a unique solution for every possible \(\mathbf{b}\).
4. ⭕ The equation \( A\mathbf{x} = \mathbf{0} \) has a nontrivial solution and equation \( A\mathbf{x} = \mathbf{b} \) does not have a unique solution for every possible \(\mathbf{b}\).
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