Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Educational Content: Understanding the Null Space and Rank-Nullity Theorem**
**Objective:**
Determine the null space of matrix \( A \) and verify the Rank-Nullity Theorem.
**Matrix \( A \):**
\[
A = \begin{bmatrix}
1 & -4 & 9 & -7 \\
-1 & 2 & -4 & 1 \\
5 & -6 & 10 & 7
\end{bmatrix}
\]
**Steps to Determine Null Space:**
1. **Gaussian Elimination:** Simplify the matrix \( A \) to its row echelon form.
2. **Solutions to Homogeneous Equation:** Solve \( A \mathbf{x} = \mathbf{0} \) to find the null space vectors.
3. **Identify Free Variables:** The number of free variables will determine the basis vectors for the null space.
**Verification of Rank-Nullity Theorem:**
- **Rank-Nullity Theorem Statement:** For a matrix \( A \) of size \( m \times n \), the sum of the rank and the nullity (dimension of null space) equals the number of columns:
\[
\text{rank}(A) + \text{nullity}(A) = n
\]
- **Application:** Calculate both the rank and nullity of \( A \) and check the above equation holds true.
This structured approach will ensure a clear understanding of determining the null space and verifying the Rank-Nullity Theorem for the given matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7df2de5-8b72-4ee1-bbed-7f478673d98d%2F3ed51edc-9da2-42dd-bc08-b3b791a22047%2Fy29ri7a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Understanding the Null Space and Rank-Nullity Theorem**
**Objective:**
Determine the null space of matrix \( A \) and verify the Rank-Nullity Theorem.
**Matrix \( A \):**
\[
A = \begin{bmatrix}
1 & -4 & 9 & -7 \\
-1 & 2 & -4 & 1 \\
5 & -6 & 10 & 7
\end{bmatrix}
\]
**Steps to Determine Null Space:**
1. **Gaussian Elimination:** Simplify the matrix \( A \) to its row echelon form.
2. **Solutions to Homogeneous Equation:** Solve \( A \mathbf{x} = \mathbf{0} \) to find the null space vectors.
3. **Identify Free Variables:** The number of free variables will determine the basis vectors for the null space.
**Verification of Rank-Nullity Theorem:**
- **Rank-Nullity Theorem Statement:** For a matrix \( A \) of size \( m \times n \), the sum of the rank and the nullity (dimension of null space) equals the number of columns:
\[
\text{rank}(A) + \text{nullity}(A) = n
\]
- **Application:** Calculate both the rank and nullity of \( A \) and check the above equation holds true.
This structured approach will ensure a clear understanding of determining the null space and verifying the Rank-Nullity Theorem for the given matrix.
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