Consider now the transformation T: R² → R² such that a stretch factor of 2 corresponds with the line y = − ½r are doubled, and a stretch factor of -1 corresponds with the line y = 4x . i. Determine what happens to the vector [] = [0,5] under this transformation and represent the answer according to the red basis, R. Explain all steps in your solution approach. ii. Determine what happens to the vector [28 = [3³] under this transformation and represent the answer according to the black basis, B. Explain all steps in your solution approach.
Consider now the transformation T: R² → R² such that a stretch factor of 2 corresponds with the line y = − ½r are doubled, and a stretch factor of -1 corresponds with the line y = 4x . i. Determine what happens to the vector [] = [0,5] under this transformation and represent the answer according to the red basis, R. Explain all steps in your solution approach. ii. Determine what happens to the vector [28 = [3³] under this transformation and represent the answer according to the black basis, B. Explain all steps in your solution approach.
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
Transcribed Image Text:**Transcription for Educational Website:**
Consider the following plane with both the red and black coordinate systems. The red axes correspond with what are normally thought of as the lines \( y = 4x \) and \( y = -\frac{1}{2}x \). Let the two red vectors highlighted along the red axis be the basis vectors for a new “red basis,” called \( \mathbf{R} = \left\{ \begin{bmatrix} 2 \\ -1 \end{bmatrix} , \begin{bmatrix} 1 \\ 4 \end{bmatrix} \right\} \). Call the standard basis \( \mathbf{B} \).
**Diagram Explanation:**
The diagram shows a coordinate plane with two sets of axes:
1. **Black Axes:** The standard coordinate system.
2. **Red Axes:** These represent new axes where the lines \( y = 4x \) and \( y = -\frac{1}{2}x \) act as the new coordinate lines.
The plane features:
- Vector lines and grid lines to illustrate the transformation between the standard black basis and the new red basis.
- Blue points labeled \( a, b, c, d, e, f \) are marked on the grid, situated at various intersections of the grid lines.
The red basis vectors are indicated prominently along the red axes. The grid lines help visualize the transformation of coordinates in the new basis system.
![Consider now the transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) such that a stretch factor of 2 corresponds with the line \( y = -\frac{1}{2}x \) are doubled, and a stretch factor of -1 corresponds with the line \( y = 4x \).
i. Determine what happens to the vector \([x]_R = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix}\) under this transformation and represent the answer according to the red basis, \( R \). Explain all steps in your solution approach.
ii. Determine what happens to the vector \([x]_B = \begin{bmatrix} -3 \\ 3 \end{bmatrix}\) under this transformation and represent the answer according to the black basis, \( B \). Explain all steps in your solution approach.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67ad69fc-d7b2-4082-8b18-7ae4b62544b8%2F66cb40af-ff15-428d-b7b7-4b85566b9b40%2Fs25u7za_processed.png&w=3840&q=75)
Transcribed Image Text:Consider now the transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) such that a stretch factor of 2 corresponds with the line \( y = -\frac{1}{2}x \) are doubled, and a stretch factor of -1 corresponds with the line \( y = 4x \).
i. Determine what happens to the vector \([x]_R = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix}\) under this transformation and represent the answer according to the red basis, \( R \). Explain all steps in your solution approach.
ii. Determine what happens to the vector \([x]_B = \begin{bmatrix} -3 \\ 3 \end{bmatrix}\) under this transformation and represent the answer according to the black basis, \( B \). Explain all steps in your solution approach.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 43 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
