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(iv) V₁ = {(₁, ₂,..., In) € R¹ | I₁I₂ = 0}. (v) V₁ = {(₁, ₂,..., n) = R¹ | 1₂ € Q}. (vi) V6 = {(₁, 2,..., In) = R¹ | x₂ € Q}.

(iv) V₁ = {(₁, ₂,..., In) € R¹ | I₁I₂ = 0}. (v) V₁ = {(₁, ₂,..., n) = R¹ | 1₂ € Q}. (vi) V6 = {(₁, 2,..., In) = R¹ | x₂ € Q}.

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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section (iv)~(vi)

**Problem 4.** Let \( n \geq 3 \). Identify which of the following subsets are subspaces of the \(\mathbb{R}\)-vector space \(\mathbb{R}^n\). Give brief justifications for your answers. (i) \( V_1 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1 \geq 0 \} \). (ii) \( V_2 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1 + 3x_2 = x_3 \} \). (iii) \( V_3 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1^2 + 9x_2^2 = x_3^2 \} \). (iv) \( V_4 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1 x_2 = 0 \} \). (v) \( V_5 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_2 \in \mathbb{Q} \} \). (vi) \( V_6 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid \pi x_2 \in \mathbb{Q} \} \).
Transcribed Image Text:**Problem 4.** Let \( n \geq 3 \). Identify which of the following subsets are subspaces of the \(\mathbb{R}\)-vector space \(\mathbb{R}^n\). Give brief justifications for your answers. (i) \( V_1 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1 \geq 0 \} \). (ii) \( V_2 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1 + 3x_2 = x_3 \} \). (iii) \( V_3 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1^2 + 9x_2^2 = x_3^2 \} \). (iv) \( V_4 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_1 x_2 = 0 \} \). (v) \( V_5 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid x_2 \in \mathbb{Q} \} \). (vi) \( V_6 = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n \mid \pi x_2 \in \mathbb{Q} \} \).
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