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Determine whether the following sets are subspaces of R³. Realize number 4 as the span of a collection of vectors. Justify your answers. 1. W₁ = = 2. W₂ = = 3. W3 = = a1 {}] a₂ € R³: a₁ = 3a2 and a3 = -A₂ -93) a1 a2 a3 a1 E a2 a3 € R³: : a₁ = a3 + 2 +9₁=0} E R³2a1 - 7a2 + a3

Determine whether the following sets are subspaces of R³. Realize number 4 as the span of a collection of vectors. Justify your answers. 1. W₁ = = 2. W₂ = = 3. W3 = = a1 {}] a₂ € R³: a₁ = 3a2 and a3 = -A₂ -93) a1 a2 a3 a1 E a2 a3 € R³: : a₁ = a3 + 2 +9₁=0} E R³2a1 - 7a2 + a3

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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### Subspaces of \( \mathbb{R}^3 \) Determine whether the following sets are subspaces of \( \mathbb{R}^3 \). Realize number 4 as the span of a collection of vectors. Justify your answers. 1. \( W_1 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 = 3a_2 \text{ and } a_3 = -a_2 \right\} \) 2. \( W_2 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 = a_3 + 2 \right\} \) 3. \( W_3 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : 2a_1 - 7a_2 + a_3 = 0 \right\} \) 4. \( W_4 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 - 4a_2 - a_3 = 0 \right\} \) 5. \( W_5 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 + 2a_2 - 3a_3 = 1 \right\} \) 6. \( W_6 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : 5a_1^2 - 3a_2^2 + 6a_3^2 = 0 \right\} \)
Transcribed Image Text:### Subspaces of \( \mathbb{R}^3 \) Determine whether the following sets are subspaces of \( \mathbb{R}^3 \). Realize number 4 as the span of a collection of vectors. Justify your answers. 1. \( W_1 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 = 3a_2 \text{ and } a_3 = -a_2 \right\} \) 2. \( W_2 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 = a_3 + 2 \right\} \) 3. \( W_3 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : 2a_1 - 7a_2 + a_3 = 0 \right\} \) 4. \( W_4 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 - 4a_2 - a_3 = 0 \right\} \) 5. \( W_5 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 + 2a_2 - 3a_3 = 1 \right\} \) 6. \( W_6 = \left\{ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \in \mathbb{R}^3 : 5a_1^2 - 3a_2^2 + 6a_3^2 = 0 \right\} \)
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