Suppose a 4x5 matrix A has two pivot columns. What is nullity A? Is Col A = R²? Why or why not? nullity A = (Simplify your answer.)
Suppose a 4x5 matrix A has two pivot columns. What is nullity A? Is Col A = R²? Why or why not? nullity A = (Simplify your answer.)
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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![**Question**: Suppose a \(4 \times 5\) matrix \(A\) has two pivot columns. What is the nullity of \(A\)? Is \(\text{Col} \, A = \mathbb{R}^2\)? Why or why not?
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**Answer**:
**Nullity \(A\)**: [Box for answer] (Simplify your answer.)
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**Explanation**:
- The **nullity** of a matrix is the dimension of the null space, which is the number of free variables. This is calculated as the number of columns minus the rank of the matrix. Given 5 columns and 2 pivot columns (rank), the nullity is \(5 - 2 = 3\).
- To determine if \(\text{Col} \, A = \mathbb{R}^2\), we consider the column space \(\text{Col} \, A\), which is defined by the number of pivot columns. Here, there are two pivot columns, meaning the column space is two-dimensional, i.e., \(\text{Col} \, A = \mathbb{R}^2\).
Thus, the column space is indeed \(\mathbb{R}^2\) because it has the same dimension.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2432691a-c29a-4a02-b85a-93b716867369%2F9ab0e9f0-5c43-4107-a626-41507417aa2a%2Ftomxism_processed.png&w=3840&q=75)
Transcribed Image Text:**Question**: Suppose a \(4 \times 5\) matrix \(A\) has two pivot columns. What is the nullity of \(A\)? Is \(\text{Col} \, A = \mathbb{R}^2\)? Why or why not?
---
**Answer**:
**Nullity \(A\)**: [Box for answer] (Simplify your answer.)
---
**Explanation**:
- The **nullity** of a matrix is the dimension of the null space, which is the number of free variables. This is calculated as the number of columns minus the rank of the matrix. Given 5 columns and 2 pivot columns (rank), the nullity is \(5 - 2 = 3\).
- To determine if \(\text{Col} \, A = \mathbb{R}^2\), we consider the column space \(\text{Col} \, A\), which is defined by the number of pivot columns. Here, there are two pivot columns, meaning the column space is two-dimensional, i.e., \(\text{Col} \, A = \mathbb{R}^2\).
Thus, the column space is indeed \(\mathbb{R}^2\) because it has the same dimension.
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