Suppose that T: R → Rm is defined by T(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(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
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Complete (c) for each transformation
![### Linear Transformations and Matrices
Suppose that \( T : \mathbb{R}^n \to \mathbb{R}^m \) is defined by \( T(\vec{x}) = A\vec{x} \) for each of the following matrices below:
\[ D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0.5 \end{bmatrix} \]
\[ E = \begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & 2 \end{bmatrix} \]
\[ F = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 3 & 0 \end{bmatrix} \]
#### (a) Rewrite \( T : \mathbb{R}^n \to \mathbb{R}^m \) 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 \( \mathbb{R}^n \) to \( \mathbb{R}^m \). You might find it 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 \( \mathbb{R}^m \))
- i. If so, explain why you can get any vector in \( \mathbb{R}^m \).
- ii. If not, give an example of an output vector you can’t get with the transformation and explain why.
### Explanation of the Matrices:
- **Matrix D**: A \(3 \times 3\) matrix that scales:
- The first component by 3.
- The second component by -1.
- The third component by 0.5.
- **Matrix E**: A \(3 \times 2\) matrix that transforms:
- The first component by 4.
- The third component by 2.
- Ignoring the input for the second component.
- **Matrix F**: A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F992f5037-f1cc-4483-8aa3-731b184958ad%2Fa7d35dae-615d-4ba8-b723-c1c93c152e6d%2Fyhe617p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Transformations and Matrices
Suppose that \( T : \mathbb{R}^n \to \mathbb{R}^m \) is defined by \( T(\vec{x}) = A\vec{x} \) for each of the following matrices below:
\[ D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0.5 \end{bmatrix} \]
\[ E = \begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & 2 \end{bmatrix} \]
\[ F = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 3 & 0 \end{bmatrix} \]
#### (a) Rewrite \( T : \mathbb{R}^n \to \mathbb{R}^m \) 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 \( \mathbb{R}^n \) to \( \mathbb{R}^m \). You might find it 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 \( \mathbb{R}^m \))
- i. If so, explain why you can get any vector in \( \mathbb{R}^m \).
- ii. If not, give an example of an output vector you can’t get with the transformation and explain why.
### Explanation of the Matrices:
- **Matrix D**: A \(3 \times 3\) matrix that scales:
- The first component by 3.
- The second component by -1.
- The third component by 0.5.
- **Matrix E**: A \(3 \times 2\) matrix that transforms:
- The first component by 4.
- The third component by 2.
- Ignoring the input for the second component.
- **Matrix F**: A
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