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-2), V1 in R³. - 40. If p₁(x) = x − 4 and p2(x) = x² − x + 3, determine whether p(x) = 2x² - x + 2 lies in span{p₁, P2}. 41. Consider the vectors 1, 2), V2 A1 A₁ = 30 4-[2-1)]-*-[-2] -4 - [39] A2 A₂ = A3 = 12 01

-2), V1 in R³. - 40. If p₁(x) = x − 4 and p2(x) = x² − x + 3, determine whether p(x) = 2x² - x + 2 lies in span{p₁, P2}. 41. Consider the vectors 1, 2), V2 A1 A₁ = 30 4-[2-1)]-*-[-2] -4 - [39] A2 A₂ = A3 = 12 01

Advanced Engineering Mathematics
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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Solve #40, SHOW every step of your work and explain. DO NOT TYPED IT, POST PICTURES OF YOUR WORK!

**Exercise Tasks:** 25. Let \( S \) be the subspace of \( M_2(\mathbb{R}) \) consisting of all \( 2 \times 2 \) matrices whose four elements sum to zero. Find a set of vectors that spans \( S \). 26. Let \( S \) be the subspace of \( M_3(\mathbb{R}) \) consisting of all \( 3 \times 3 \) matrices such that the elements in each row and each column sum to zero. Find a set of vectors that spans \( S \). 27. Let \( S \) be the subspace of \( M_3(\mathbb{R}) \) consisting of all \( 3 \times 3 \) skew-symmetric matrices. Find a set of vectors that spans \( S \). 28. Let \( S \) be the subspace of \( \mathbb{R}^3 \) consisting of all solutions to the linear system \( x - 2y - z = 0 \). Determine a set of vectors that spans \( S \). 29. 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 \( S \). **Determine whether the given vector lies in a span:** For Problems 37–39, determine whether the given vector \( \mathbf{v} \) lies in \( \text{span}\{\mathbf{v_1}, \mathbf{v_2}\} \). 37. \( \mathbf{v} = (3, 3, 4), \, \mathbf{v_1} = (1, -1, 2), \, \mathbf{v_2} = (2, 1, 3) \) in \( \mathbb{R}^3 \). 38. \( \mathbf{v} = (5, 3, -6), \, \mathbf{v_1} = (-1, 1, 2), \, \mathbf{v_2} = (3, 1, -4) \) in \( \mathbb{R}^3 \). 39. \(
Transcribed Image Text:**Exercise Tasks:** 25. Let \( S \) be the subspace of \( M_2(\mathbb{R}) \) consisting of all \( 2 \times 2 \) matrices whose four elements sum to zero. Find a set of vectors that spans \( S \). 26. Let \( S \) be the subspace of \( M_3(\mathbb{R}) \) consisting of all \( 3 \times 3 \) matrices such that the elements in each row and each column sum to zero. Find a set of vectors that spans \( S \). 27. Let \( S \) be the subspace of \( M_3(\mathbb{R}) \) consisting of all \( 3 \times 3 \) skew-symmetric matrices. Find a set of vectors that spans \( S \). 28. Let \( S \) be the subspace of \( \mathbb{R}^3 \) consisting of all solutions to the linear system \( x - 2y - z = 0 \). Determine a set of vectors that spans \( S \). 29. 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 \( S \). **Determine whether the given vector lies in a span:** For Problems 37–39, determine whether the given vector \( \mathbf{v} \) lies in \( \text{span}\{\mathbf{v_1}, \mathbf{v_2}\} \). 37. \( \mathbf{v} = (3, 3, 4), \, \mathbf{v_1} = (1, -1, 2), \, \mathbf{v_2} = (2, 1, 3) \) in \( \mathbb{R}^3 \). 38. \( \mathbf{v} = (5, 3, -6), \, \mathbf{v_1} = (-1, 1, 2), \, \mathbf{v_2} = (3, 1, -4) \) in \( \mathbb{R}^3 \). 39. \(
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