Let T: UV be a linear transformation. Use the rank-nullity theorem t complete the information in the table below. U dim(U) rank (T) nullity (T) R1x3 3 2 Ex: 5 R2x2 Ex: 5 Ex: 5 3 R7x4 Ex: 5 Ex: 5 4
Let T: UV be a linear transformation. Use the rank-nullity theorem t complete the information in the table below. U dim(U) rank (T) nullity (T) R1x3 3 2 Ex: 5 R2x2 Ex: 5 Ex: 5 3 R7x4 Ex: 5 Ex: 5 4
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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![**Linear Transformation and the Rank-Nullity Theorem**
Let \( T : U \to V \) be a linear transformation. Use the rank-nullity theorem to complete the information in the table below.
| \( U \) | \( \mathbb{R}_{1 \times 3} \) | \( \mathbb{R}_{2 \times 2} \) | \( \mathbb{R}_{7 \times 4} \) |
|--------------|----------------|----------------|----------------|
| \(\dim(U)\) | 3 | Ex: 5 | Ex: 5 |
| \(\text{rank}(T)\) | 2 | Ex: 5 | Ex: 5 |
| \(\text{nullity}(T)\) | Ex: 5 | 3 | 4 |
**Explanation:**
- **\(\dim(U)\)**: Represents the dimension of the vector space \( U \).
- **\(\text{rank}(T)\)**: Represents the rank of the linear transformation \( T \).
- **\(\text{nullity}(T)\)**: Represents the nullity of the linear transformation \( T \).
The Rank-Nullity Theorem states:
\[
\dim(U) = \text{rank}(T) + \text{nullity}(T)
\]
Use this theorem to solve for the missing values in the table as labeled "Ex: 5."](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97fa71a9-ddb9-496b-9b0a-bf970e388fad%2Faec6c750-6c97-4718-9b71-634567b68500%2F0idaxsg_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Transformation and the Rank-Nullity Theorem**
Let \( T : U \to V \) be a linear transformation. Use the rank-nullity theorem to complete the information in the table below.
| \( U \) | \( \mathbb{R}_{1 \times 3} \) | \( \mathbb{R}_{2 \times 2} \) | \( \mathbb{R}_{7 \times 4} \) |
|--------------|----------------|----------------|----------------|
| \(\dim(U)\) | 3 | Ex: 5 | Ex: 5 |
| \(\text{rank}(T)\) | 2 | Ex: 5 | Ex: 5 |
| \(\text{nullity}(T)\) | Ex: 5 | 3 | 4 |
**Explanation:**
- **\(\dim(U)\)**: Represents the dimension of the vector space \( U \).
- **\(\text{rank}(T)\)**: Represents the rank of the linear transformation \( T \).
- **\(\text{nullity}(T)\)**: Represents the nullity of the linear transformation \( T \).
The Rank-Nullity Theorem states:
\[
\dim(U) = \text{rank}(T) + \text{nullity}(T)
\]
Use this theorem to solve for the missing values in the table as labeled "Ex: 5."
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