Suppose that T: R" → Rm is defined by T() = A for each of the following matrices below: [3 0 0 D = #) 449 0 -1 0 0 F = 0 3 0 0 0.5 E = 20 (a) Rewrite T: RnRm with correct numbers for m and n for each transformation. What is the domain and codomain of each transformation? (b) Find some way to explain in words and/or graphically what each transformation does as it takes vectors from R" to Rm. You might find it t helpful to try out a few input vectors and see what their image is under the transformation. This might be difficult, but an honest effort will give you credit. (c) For the transformation, can you get any output vector? (Any vector in Rm) i. If so, explain why you can get any vector in Rm. ii. If not, give an example of an output vector you can't get with the transformation and explain why.
Suppose that T: R" → Rm is defined by T() = A for each of the following matrices below: [3 0 0 D = #) 449 0 -1 0 0 F = 0 3 0 0 0.5 E = 20 (a) Rewrite T: RnRm with correct numbers for m and n for each transformation. What is the domain and codomain of each transformation? (b) Find some way to explain in words and/or graphically what each transformation does as it takes vectors from R" to Rm. You might find it t helpful to try out a few input vectors and see what their image is under the transformation. This might be difficult, but an honest effort will give you credit. (c) For the transformation, can you get any output vector? (Any vector in Rm) i. If so, explain why you can get any vector in Rm. ii. If not, give an example of an output vector you can't get with the transformation and explain why.
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
Complete (b) for each transformation
![**Linear Transformations and Matrices**
Suppose that \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is defined by \( T(\vec{x}) = A\vec{x} \) for each of the following matrices below:
\[
D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0.5 \end{bmatrix} \quad
E = \begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & 2 \end{bmatrix} \quad
F = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 3 & 0 \end{bmatrix}
\]
**Tasks:**
(a) **Rewrite \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) with correct numbers for \( m \) and \( n \) for each transformation. What is the domain and codomain of each transformation?**
- **Matrix \( D \):** Since \( D \) is a 3x3 matrix, \( n = 3 \) and \( m = 3 \). Thus, \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \).
- **Matrix \( E \):** Since \( E \) is a 3x2 matrix, \( n = 2 \) and \( m = 3 \). Thus, \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \).
- **Matrix \( F \):** Since \( F \) is a 2x3 matrix, \( n = 3 \) and \( m = 2 \). Thus, \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \).
(b) **Find some way to explain in words and/or graphically what each transformation does as it takes vectors from \( \mathbb{R}^n \) to \( \mathbb{R}^m \).**
- **Matrix \( D \):** This transformation scales the first component by 3, the second by -1 (reflection), and the third by 0.5, leaving the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F992f5037-f1cc-4483-8aa3-731b184958ad%2F40b00565-ce56-4cda-ae0b-164a7f048702%2F5uqrde_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Transformations and Matrices**
Suppose that \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) is defined by \( T(\vec{x}) = A\vec{x} \) for each of the following matrices below:
\[
D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0.5 \end{bmatrix} \quad
E = \begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & 2 \end{bmatrix} \quad
F = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 3 & 0 \end{bmatrix}
\]
**Tasks:**
(a) **Rewrite \( T : \mathbb{R}^n \rightarrow \mathbb{R}^m \) with correct numbers for \( m \) and \( n \) for each transformation. What is the domain and codomain of each transformation?**
- **Matrix \( D \):** Since \( D \) is a 3x3 matrix, \( n = 3 \) and \( m = 3 \). Thus, \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \).
- **Matrix \( E \):** Since \( E \) is a 3x2 matrix, \( n = 2 \) and \( m = 3 \). Thus, \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \).
- **Matrix \( F \):** Since \( F \) is a 2x3 matrix, \( n = 3 \) and \( m = 2 \). Thus, \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \).
(b) **Find some way to explain in words and/or graphically what each transformation does as it takes vectors from \( \mathbb{R}^n \) to \( \mathbb{R}^m \).**
- **Matrix \( D \):** This transformation scales the first component by 3, the second by -1 (reflection), and the third by 0.5, leaving the
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 3 steps with 2 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,
