3. If a linear transformation T : R³ → R² has a 2 dimensional kernel, what is the dimension of the range? A. 0 В. 1 С. 2 D. Can't be found from the information given
3. If a linear transformation T : R³ → R² has a 2 dimensional kernel, what is the dimension of the range? A. 0 В. 1 С. 2 D. Can't be found from the information given
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
Multiple Choice
![**Title: Quiz on Linear Transformations and Vector Dependence**
**Instruction: Answer the following multiple-choice questions related to linear algebra topics. Review any concepts you are unsure about before making your selection.**
---
**Question 3:**
If a linear transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) has a 2-dimensional kernel, what is the dimension of the range?
- **A.** 0
- **B.** 1
- **C.** 2
- **D.** Can't be found from the information given
---
**Question 4:**
Which of the following sets of vectors in \(\mathbb{R}^2\) is linearly dependent?
- **A.** \(\{(1, 0), (0, 1)\}\)
- **B.** \(\{(1, 0), (0, 1), (1, 3)\}\)
- **C.** \(\{(1, 0), (1, 3)\}\)
- **D.** \(\{(0, 1), (1, 3)\}\)
---
**Explanation of Concepts:**
- **Linear Transformation and Kernel:**
A linear transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) maps vectors from a 3-dimensional space to a 2-dimensional space. The kernel (or null space) of \( T \) is the set of all vectors in \(\mathbb{R}^3\) that \( T \) maps to the zero vector in \(\mathbb{R}^2\). The dimension of the kernel, also known as the nullity, plays a role in determining the dimension of the range (or image) via the Rank-Nullity Theorem.
- **Rank-Nullity Theorem:**
The Rank-Nullity Theorem states that for a linear transformation \( T \) from \(\mathbb{R}^n \) to \(\mathbb{R}^m \):
\[
\text{dim}(\text{Domain}) = \text{Rank}(T) + \text{Nullity}(T)
\]
In this case, \(\text{dim}(\text{Domain}) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f133c63-1716-41d7-8ec5-0f87428512d1%2F597938c8-4323-43ee-bf28-4ccbfce246d0%2F2ioctq_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Quiz on Linear Transformations and Vector Dependence**
**Instruction: Answer the following multiple-choice questions related to linear algebra topics. Review any concepts you are unsure about before making your selection.**
---
**Question 3:**
If a linear transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) has a 2-dimensional kernel, what is the dimension of the range?
- **A.** 0
- **B.** 1
- **C.** 2
- **D.** Can't be found from the information given
---
**Question 4:**
Which of the following sets of vectors in \(\mathbb{R}^2\) is linearly dependent?
- **A.** \(\{(1, 0), (0, 1)\}\)
- **B.** \(\{(1, 0), (0, 1), (1, 3)\}\)
- **C.** \(\{(1, 0), (1, 3)\}\)
- **D.** \(\{(0, 1), (1, 3)\}\)
---
**Explanation of Concepts:**
- **Linear Transformation and Kernel:**
A linear transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) maps vectors from a 3-dimensional space to a 2-dimensional space. The kernel (or null space) of \( T \) is the set of all vectors in \(\mathbb{R}^3\) that \( T \) maps to the zero vector in \(\mathbb{R}^2\). The dimension of the kernel, also known as the nullity, plays a role in determining the dimension of the range (or image) via the Rank-Nullity Theorem.
- **Rank-Nullity Theorem:**
The Rank-Nullity Theorem states that for a linear transformation \( T \) from \(\mathbb{R}^n \) to \(\mathbb{R}^m \):
\[
\text{dim}(\text{Domain}) = \text{Rank}(T) + \text{Nullity}(T)
\]
In this case, \(\text{dim}(\text{Domain}) =
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