Suppose B is a 5 x 5 matrix and Nul B is NOT the zero subspace. Do you have enough information to determine if Col B equals R5? Why or why not?
Suppose B is a 5 x 5 matrix and Nul B is NOT the zero subspace. Do you have enough information to determine if Col B equals R5? Why or why not?
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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![**Title: Understanding Column Spaces in Linear Algebra**
**Problem:**
Suppose \( B \) is a \( 5 \times 5 \) matrix and \(\text{Nul } B\) is NOT the zero subspace. Do you have enough information to determine if \(\text{Col } B\) equals \(\mathbb{R}^5\)? Why or why not?
**Explanation:**
To determine whether the column space of the matrix \( B \) (denoted as \(\text{Col } B\)) equals \(\mathbb{R}^5\), we need to assess the concept of linear independence and the rank of the matrix \( B \).
- **Key Concept: Rank-Nullity Theorem**
The rank-nullity theorem states that for any matrix \( A \) of size \( m \times n \), the sum of the rank and the nullity (dimension of the null space) equals \( n\):
\[
\text{rank}(A) + \text{nullity}(A) = n
\]
- **Implication for the 5x5 Matrix \( B \)**
Given that \(\text{Nul } B\) is NOT the zero subspace, it means that the nullity of \( B \) is greater than 0. Thus, the rank of \( B \) is less than 5.
- **Effect on Column Space**
For \(\text{Col } B\) to equal \(\mathbb{R}^5\), the rank of \( B \) must be 5. However, because the nullity is greater than 0, the rank must be less than 5. Therefore, \(\text{Col } B\) cannot equal \(\mathbb{R}^5\).
**Conclusion:**
No, we do not have enough information to say \(\text{Col } B\) equals \(\mathbb{R}^5\). In fact, with the given condition that \(\text{Nul } B\) is NOT the zero subspace, we can definitively conclude that \(\text{Col } B\) does not, and cannot, equal \(\mathbb{R}^5\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30f79184-3047-455a-b3d8-f2ad47623cdf%2Fc6f5c0ca-274d-40e7-9262-b2683a07841e%2Fac2ief9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Column Spaces in Linear Algebra**
**Problem:**
Suppose \( B \) is a \( 5 \times 5 \) matrix and \(\text{Nul } B\) is NOT the zero subspace. Do you have enough information to determine if \(\text{Col } B\) equals \(\mathbb{R}^5\)? Why or why not?
**Explanation:**
To determine whether the column space of the matrix \( B \) (denoted as \(\text{Col } B\)) equals \(\mathbb{R}^5\), we need to assess the concept of linear independence and the rank of the matrix \( B \).
- **Key Concept: Rank-Nullity Theorem**
The rank-nullity theorem states that for any matrix \( A \) of size \( m \times n \), the sum of the rank and the nullity (dimension of the null space) equals \( n\):
\[
\text{rank}(A) + \text{nullity}(A) = n
\]
- **Implication for the 5x5 Matrix \( B \)**
Given that \(\text{Nul } B\) is NOT the zero subspace, it means that the nullity of \( B \) is greater than 0. Thus, the rank of \( B \) is less than 5.
- **Effect on Column Space**
For \(\text{Col } B\) to equal \(\mathbb{R}^5\), the rank of \( B \) must be 5. However, because the nullity is greater than 0, the rank must be less than 5. Therefore, \(\text{Col } B\) cannot equal \(\mathbb{R}^5\).
**Conclusion:**
No, we do not have enough information to say \(\text{Col } B\) equals \(\mathbb{R}^5\). In fact, with the given condition that \(\text{Nul } B\) is NOT the zero subspace, we can definitively conclude that \(\text{Col } B\) does not, and cannot, equal \(\mathbb{R}^5\).
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