For Problems 25-33, determine a spanning set for the null space of the given matrix A.
For Problems 25-33, determine a spanning set for the null space of the given matrix A.
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
solve number 29 of the first picture using the second picture you are able to see the matrix which is problem #27 of the second picture.
![Certainly! Here is a transcription and explanation suitable for an educational website:
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### Mathematics: Subspaces and Null Spaces
Explore various vector spaces \( V \) and subsets \( S \) with different conditions, and determine the null spaces for a range of matrices.
**Subspaces Descriptions:**
1. **Problem 8:**
- \( V = \mathbb{R}^2 \), and \( S \) consists of vectors \((x, y)\) satisfying \( x^2 - y^2 = 0 \).
2. **Problem 9:**
- \( V = M_2(\mathbb{R}) \), and \( S \) is the subset of all \( 2 \times 2 \) matrices with \(\det(A) = 1\).
3. **Problem 10:**
- \( V = M_n(\mathbb{R}) \), and \( S \) is the subset of all \( n \times n \) lower triangular matrices.
4. **Problem 11:**
- \( V = M_n(\mathbb{R}) \), and \( S \) is the subset of all \( n \times n \) invertible matrices.
5. **Problem 12:**
- \( V = M_2(\mathbb{R}) \), and \( S \) is the subset where all four elements sum to zero.
6. **Problem 13:**
- \( V = M_{3 \times 2}(\mathbb{R}) \), and \( S \) is the subset where elements in each column sum to zero.
7. **Problem 14:**
- \( V = M_{2 \times 3}(\mathbb{R}) \), and \( S \) is the subset with each row summing to 10.
8. **Problem 15:**
- \( V = M_2(\mathbb{R}) \), and \( S \) is the set of all \( 2 \times 2 \) symmetric matrices.
9. **Problem 16:**
- \( V \) includes all real-valued functions on \([a, b]\), with \( S \) being functions satisfying \( f(a) = 5 \cdot f(b) \).
10. **Problem 17:**
- \( V \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2Ff1cd3ca2-f5b9-4e02-a404-dfe29de775d4%2Fo473db_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Here is a transcription and explanation suitable for an educational website:
---
### Mathematics: Subspaces and Null Spaces
Explore various vector spaces \( V \) and subsets \( S \) with different conditions, and determine the null spaces for a range of matrices.
**Subspaces Descriptions:**
1. **Problem 8:**
- \( V = \mathbb{R}^2 \), and \( S \) consists of vectors \((x, y)\) satisfying \( x^2 - y^2 = 0 \).
2. **Problem 9:**
- \( V = M_2(\mathbb{R}) \), and \( S \) is the subset of all \( 2 \times 2 \) matrices with \(\det(A) = 1\).
3. **Problem 10:**
- \( V = M_n(\mathbb{R}) \), and \( S \) is the subset of all \( n \times n \) lower triangular matrices.
4. **Problem 11:**
- \( V = M_n(\mathbb{R}) \), and \( S \) is the subset of all \( n \times n \) invertible matrices.
5. **Problem 12:**
- \( V = M_2(\mathbb{R}) \), and \( S \) is the subset where all four elements sum to zero.
6. **Problem 13:**
- \( V = M_{3 \times 2}(\mathbb{R}) \), and \( S \) is the subset where elements in each column sum to zero.
7. **Problem 14:**
- \( V = M_{2 \times 3}(\mathbb{R}) \), and \( S \) is the subset with each row summing to 10.
8. **Problem 15:**
- \( V = M_2(\mathbb{R}) \), and \( S \) is the set of all \( 2 \times 2 \) symmetric matrices.
9. **Problem 16:**
- \( V \) includes all real-valued functions on \([a, b]\), with \( S \) being functions satisfying \( f(a) = 5 \cdot f(b) \).
10. **Problem 17:**
- \( V \)
![**Linear Algebra Problem Set**
**23.** Solve the linear system:
\[ x - 2y - z = 0. \]
Determine a set of vectors that spans \( S \).
**24.** Let \( S \) be the subspace of \( P_3(\mathbb{R}) \) consisting of all polynomials \( p(x) \) in \( P_3(\mathbb{R}) \) such that \( p'(x) = 0 \). Find a set of vectors that spans \( S \).
For Problems **25–33**, determine a spanning set for the null space of the given matrix \( A \).
**25.** The matrix \( A \) defined in Problem 23 in Section 4.3.
**26.** The matrix \( A \) defined in Problem 24 in Section 4.3.
**27.** The matrix \( A \) defined in Problem 25 in Section 4.3.
**28.** The matrix \( A \) defined in Problem 26 in Section 4.3.
**29.** The matrix \( A \) defined in Problem 27 in Section 4.3.
**30.** The matrix \( A \) defined in Problem 28 in Section 4.3.
**31.** The matrix \( A \) defined in Problem 29 in Section 4.3.
**32.** Given matrix:
\[ A = \begin{bmatrix} 1 & 2 & 3 & 5 \\ 1 & 3 & 4 & 2 \\ 2 & 4 & 6 & -1 \end{bmatrix}. \]
**41.** Consider the vectors
\[ A_1 = \begin{bmatrix} 1 & -1 \\ 2 & 0 \end{bmatrix}, \quad A_2 = \begin{bmatrix} 0 & 1 \\ -2 & 1 \end{bmatrix}, \quad A_3 = \begin{bmatrix} 3 & 0 \\ 1 & 2 \end{bmatrix} \]
in \( M_2(\mathbb{R}) \). Determine \(\text{span}\{A_1, A_2, A_3\}\).
**42.** Consider the vectors
\[ A_1 = \begin{bmatrix}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2Ff1cd3ca2-f5b9-4e02-a404-dfe29de775d4%2Fwjbt2qi_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Problem Set**
**23.** Solve the linear system:
\[ x - 2y - z = 0. \]
Determine a set of vectors that spans \( S \).
**24.** Let \( S \) be the subspace of \( P_3(\mathbb{R}) \) consisting of all polynomials \( p(x) \) in \( P_3(\mathbb{R}) \) such that \( p'(x) = 0 \). Find a set of vectors that spans \( S \).
For Problems **25–33**, determine a spanning set for the null space of the given matrix \( A \).
**25.** The matrix \( A \) defined in Problem 23 in Section 4.3.
**26.** The matrix \( A \) defined in Problem 24 in Section 4.3.
**27.** The matrix \( A \) defined in Problem 25 in Section 4.3.
**28.** The matrix \( A \) defined in Problem 26 in Section 4.3.
**29.** The matrix \( A \) defined in Problem 27 in Section 4.3.
**30.** The matrix \( A \) defined in Problem 28 in Section 4.3.
**31.** The matrix \( A \) defined in Problem 29 in Section 4.3.
**32.** Given matrix:
\[ A = \begin{bmatrix} 1 & 2 & 3 & 5 \\ 1 & 3 & 4 & 2 \\ 2 & 4 & 6 & -1 \end{bmatrix}. \]
**41.** Consider the vectors
\[ A_1 = \begin{bmatrix} 1 & -1 \\ 2 & 0 \end{bmatrix}, \quad A_2 = \begin{bmatrix} 0 & 1 \\ -2 & 1 \end{bmatrix}, \quad A_3 = \begin{bmatrix} 3 & 0 \\ 1 & 2 \end{bmatrix} \]
in \( M_2(\mathbb{R}) \). Determine \(\text{span}\{A_1, A_2, A_3\}\).
**42.** Consider the vectors
\[ A_1 = \begin{bmatrix}
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