Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Finding the Nullity of Matrix A
**Problem Statement:**
In this problem, we need to find the dimensions of the null space (Nul A).
**Given:**
- \( A \) is a \( 4 \times 7 \) matrix and it has three pivot columns.
Clearly, \( A \) has four rows and seven columns.
\[ \text{col}(A) = \mathbb{R}^3 \]
**Explanation:**
- \( A \) has 7 columns and 3 pivot columns, so \( \text{col}(A) \) is a three-dimensional subspace of \( \mathbb{R}^4 \).
- Given \( A \) has 4 rows, \( \text{col}(A) \) is a subspace of \( \mathbb{R}^4 \), so it cannot equal \( \mathbb{R}^3 \).
The dimension of the column space of \( A \) is the rank of \( A \). Thus, \(\text{rank}(A) = 3\).
**Application of the Rank-Nullity Theorem:**
By using the Rank-Nullity Theorem:
\[ n = \text{null}(A) + \text{rank}(A) \]
Substituting the given values:
\[ \Rightarrow 7 = \text{null}(A) + 3 \]
Solving for \(\text{null}(A)\):
\[ \Rightarrow 7 - 3 = \text{null}(A) \]
Therefore, \(\text{null}(A) = 4\).
**Rank-Nullity Theorem Description:**
- Let \( A \) be an \( m \times n \) matrix,
- The theorem states: \[ n = \text{null}(A) + \text{rank}(A) \]
- \( n \) is the number of columns of \( A \).
This was the step-by-step process to find the nullity of matrix \( A \).](https://content.bartleby.com/qna-images/question/c5518ab0-f2fc-4b75-bea2-7bc0edfda8d7/b1c4e318-5b47-45cb-a1e1-da22a96653bf/h32h0dw_processed.png)
Transcribed Image Text:### Finding the Nullity of Matrix A
**Problem Statement:**
In this problem, we need to find the dimensions of the null space (Nul A).
**Given:**
- \( A \) is a \( 4 \times 7 \) matrix and it has three pivot columns.
Clearly, \( A \) has four rows and seven columns.
\[ \text{col}(A) = \mathbb{R}^3 \]
**Explanation:**
- \( A \) has 7 columns and 3 pivot columns, so \( \text{col}(A) \) is a three-dimensional subspace of \( \mathbb{R}^4 \).
- Given \( A \) has 4 rows, \( \text{col}(A) \) is a subspace of \( \mathbb{R}^4 \), so it cannot equal \( \mathbb{R}^3 \).
The dimension of the column space of \( A \) is the rank of \( A \). Thus, \(\text{rank}(A) = 3\).
**Application of the Rank-Nullity Theorem:**
By using the Rank-Nullity Theorem:
\[ n = \text{null}(A) + \text{rank}(A) \]
Substituting the given values:
\[ \Rightarrow 7 = \text{null}(A) + 3 \]
Solving for \(\text{null}(A)\):
\[ \Rightarrow 7 - 3 = \text{null}(A) \]
Therefore, \(\text{null}(A) = 4\).
**Rank-Nullity Theorem Description:**
- Let \( A \) be an \( m \times n \) matrix,
- The theorem states: \[ n = \text{null}(A) + \text{rank}(A) \]
- \( n \) is the number of columns of \( A \).
This was the step-by-step process to find the nullity of matrix \( A \).
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