Use the function to find the image of v and the preimage of w. T(V3, V2) = ( V2, V1 + V2, 2v1 - V2 2 v = (2, 2), w = (-6/2, 0, -18) (a) the image of v (0,4,2) (b) the preimage of w (If the vector has an infinite number of solutions, give your answer in terms of the parameter t.) (2,36)
Use the function to find the image of v and the preimage of w. T(V3, V2) = ( V2, V1 + V2, 2v1 - V2 2 v = (2, 2), w = (-6/2, 0, -18) (a) the image of v (0,4,2) (b) the preimage of w (If the vector has an infinite number of solutions, give your answer in terms of the parameter t.) (2,36)
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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![### Image and Preimage Calculation Using a Transformation Function
#### Transformation Function
We are provided with a transformation function \( T(v_1, v_2) \):
\[
T(v_1, v_2) = \left(\frac{\sqrt{2}}{2}v_1 - \frac{\sqrt{2}}{2}v_2, v_1 + v_2, 2v_1 - v_2 \right)
\]
#### Given Vectors
- **Vector \( \mathbf{v} \)**:
\[
\mathbf{v} = (2, 2)
\]
- **Vector \( \mathbf{w} \)**:
\[
\mathbf{w} = \left(-6\sqrt{2}, 0, -18\right)
\]
### Problem Statements
**(a) The Image of \( \mathbf{v} \)**
The image of vector \( \mathbf{v} \) under the transformation is:
\[
(0, 4, 2)
\]
This solution is verified as correct with a check mark.
**(b) The Preimage of \( \mathbf{w} \)**
The preimage of vector \( \mathbf{w} \) is:
\[
(2, 36)
\]
This solution is marked incorrect with a cross.
**Note:** If the vector has an infinite number of solutions, express your answer in terms of the parameter \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe36d4f3b-92f0-480a-8daa-f4dd6549adfe%2Fe883a8c8-93a1-4aa5-8409-13f515892556%2Fmoikdoc_processed.png&w=3840&q=75)
Transcribed Image Text:### Image and Preimage Calculation Using a Transformation Function
#### Transformation Function
We are provided with a transformation function \( T(v_1, v_2) \):
\[
T(v_1, v_2) = \left(\frac{\sqrt{2}}{2}v_1 - \frac{\sqrt{2}}{2}v_2, v_1 + v_2, 2v_1 - v_2 \right)
\]
#### Given Vectors
- **Vector \( \mathbf{v} \)**:
\[
\mathbf{v} = (2, 2)
\]
- **Vector \( \mathbf{w} \)**:
\[
\mathbf{w} = \left(-6\sqrt{2}, 0, -18\right)
\]
### Problem Statements
**(a) The Image of \( \mathbf{v} \)**
The image of vector \( \mathbf{v} \) under the transformation is:
\[
(0, 4, 2)
\]
This solution is verified as correct with a check mark.
**(b) The Preimage of \( \mathbf{w} \)**
The preimage of vector \( \mathbf{w} \) is:
\[
(2, 36)
\]
This solution is marked incorrect with a cross.
**Note:** If the vector has an infinite number of solutions, express your answer in terms of the parameter \( t \).
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