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you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false. (a) The null space of an m x n matrix A with real elements is a subspace of Rm. (b) The solution set of any linear system of m equations in n variables forms a subspace of C". (c) The points in R² that lie on the line y = mx + b form a subspace of R2 if and only if b = 0. (d) If m

you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false. (a) The null space of an m x n matrix A with real elements is a subspace of Rm. (b) The solution set of any linear system of m equations in n variables forms a subspace of C". (c) The points in R² that lie on the line y = mx + b form a subspace of R2 if and only if b = 0. (d) If m

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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or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false. (a) The null space of an m x n matrix A with real elements is a subspace of Rm. (b) The solution set of any linear system of m equations in n variables forms a subspace of C". (c) The points in R2 that lie on the line y = mx + b form a subspace of R2 if and only if b = 0. (d) If m <n, then R" is a subspace of R". Problems 1. Let S = {x € R³ : x = 2. Let S For Problems 3-22, express S in set notation and determine whether it is a subspace of the given vector space V. 3. V = R³, and S is the set of all vectors (x, y, z) in V such that z = 3x and y 2x. = 4. V = R2, and S is the set of all vectors (x, y) in V satisfying 3x + 2y = 0. 5. V = R¹, and S is the set of all vectors of the form (x1, 0, x3, 2). = (r − 2s, 3r+s, s), r, s ≤ R}. (a) Show that S is a subspace of R³. (b) Show that the vectors in S lie on the plane with equation 3x – y + 7z = 0. = {x € R² : x = (2k, −3k), k ≤ R}. (a) Show that S is a subspace of R². (b) Make a sketch depicting the subspace S in the Cartesian plane. 4.3 Subspaces 273 19. V = P₂ (R), and S is the subset of P2 (R) consisting of all polynomials of the form p(x) = ax² + b. 20. V = P₂(R), and S is the subset of P₂ (R) consisting of all polynomials of the form p(x) = ax² + 1. 21. V C²(1), and S is the subset of V consisting of those functions satisfying the differential equation y" + 2y' - y = 0
Transcribed Image Text:or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false. (a) The null space of an m x n matrix A with real elements is a subspace of Rm. (b) The solution set of any linear system of m equations in n variables forms a subspace of C". (c) The points in R2 that lie on the line y = mx + b form a subspace of R2 if and only if b = 0. (d) If m <n, then R" is a subspace of R". Problems 1. Let S = {x € R³ : x = 2. Let S For Problems 3-22, express S in set notation and determine whether it is a subspace of the given vector space V. 3. V = R³, and S is the set of all vectors (x, y, z) in V such that z = 3x and y 2x. = 4. V = R2, and S is the set of all vectors (x, y) in V satisfying 3x + 2y = 0. 5. V = R¹, and S is the set of all vectors of the form (x1, 0, x3, 2). = (r − 2s, 3r+s, s), r, s ≤ R}. (a) Show that S is a subspace of R³. (b) Show that the vectors in S lie on the plane with equation 3x – y + 7z = 0. = {x € R² : x = (2k, −3k), k ≤ R}. (a) Show that S is a subspace of R². (b) Make a sketch depicting the subspace S in the Cartesian plane. 4.3 Subspaces 273 19. V = P₂ (R), and S is the subset of P2 (R) consisting of all polynomials of the form p(x) = ax² + b. 20. V = P₂(R), and S is the subset of P₂ (R) consisting of all polynomials of the form p(x) = ax² + 1. 21. V C²(1), and S is the subset of V consisting of those functions satisfying the differential equation y" + 2y' - y = 0
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