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:---### Mathematics: Subspaces and Null SpacesExplore 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}

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Answered: For Problems 25-33, determine a…

For Problems 25-33, determine a spanning set for the null space of the given matrix A.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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