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? ? ? ? ? Determine whether the given set S is a subspace of the vector space V. 1. V = RXn, and S is the subset of all upper triangular matrices. 2. V = RXn, and S is the subset of all symmetric matrices. 3. V = R³, and S is the set of vectors (x₁, x2, x3) in V satisfying x₁ - 7x₂ + x3 = 6. 4. V = P5, and S is the subset of P5 consisting of those polynomials satisfying p(1) > p(0). 5. V = C¹(R), and S is the subset of V consisting of those functions f satisfying f'(0) ≥ 0. Notation: P₁ is the vector space of polynomials of degree up to n, and C" (R) is the vector space of n times continuously differentiable functions on R.

? ? ? ? ? Determine whether the given set S is a subspace of the vector space V. 1. V = RXn, and S is the subset of all upper triangular matrices. 2. V = RXn, and S is the subset of all symmetric matrices. 3. V = R³, and S is the set of vectors (x₁, x2, x3) in V satisfying x₁ - 7x₂ + x3 = 6. 4. V = P5, and S is the subset of P5 consisting of those polynomials satisfying p(1) > p(0). 5. V = C¹(R), and S is the subset of V consisting of those functions f satisfying f'(0) ≥ 0. Notation: P₁ is the vector space of polynomials of degree up to n, and C" (R) is the vector space of n times continuously differentiable functions on R.

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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? ? ? ? ? Determine whether the given set S is a subspace of the vector space V. 1. V = RXn, and S is the subset of all upper triangular matrices. 2. V = RXn, and S is the subset of all symmetric matrices. 3. V = R³, and S is the set of vectors (x₁, x₂, x3)T in V satisfying x₁ - 7x₂ + x3 = 6. 4. V = P5, and S is the subset of P5 consisting of those polynomials satisfying p(1) > p(0). 5. V = C¹(R), and S is the subset of V consisting of those functions f satisfying ƒ'(0) ≥ 0. Notation: Pn is the vector space of polynomials of degree up to n, and C¹(R) is the vector space of n times continuously differentiable functions on R.
Transcribed Image Text:? ? ? ? ? Determine whether the given set S is a subspace of the vector space V. 1. V = RXn, and S is the subset of all upper triangular matrices. 2. V = RXn, and S is the subset of all symmetric matrices. 3. V = R³, and S is the set of vectors (x₁, x₂, x3)T in V satisfying x₁ - 7x₂ + x3 = 6. 4. V = P5, and S is the subset of P5 consisting of those polynomials satisfying p(1) > p(0). 5. V = C¹(R), and S is the subset of V consisting of those functions f satisfying ƒ'(0) ≥ 0. Notation: Pn is the vector space of polynomials of degree up to n, and C¹(R) is the vector space of n times continuously differentiable functions on R.
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