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linear algebra 3.1 Q1 a, c, e

linear algebra 3.1 Q1 a, c, e

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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linear algebra 3.1 Q1 a, c, e

**1. Which of the following are subspaces? Justify your answer in each case.** a. \(\{ \mathbf{x} \in \mathbb{R}^2 : x_1 + x_2 = 1 \}\) b. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \begin{bmatrix} a \\ b \\ a + b \end{bmatrix} \text{ for some } a, b \in \mathbb{R} \}\) c. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1 + 2x_2 < 0 \}\) d. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \}\) e. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 0 \}\) f. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = -1 \}\) g. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = s \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + t \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} \text{ for some } s, t \in \mathbb{R} \}\) h. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} + s \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + t \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} \text{ for some } s, t \in \mathbb{R} \}\) i. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf
Transcribed Image Text:**1. Which of the following are subspaces? Justify your answer in each case.** a. \(\{ \mathbf{x} \in \mathbb{R}^2 : x_1 + x_2 = 1 \}\) b. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \begin{bmatrix} a \\ b \\ a + b \end{bmatrix} \text{ for some } a, b \in \mathbb{R} \}\) c. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1 + 2x_2 < 0 \}\) d. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \}\) e. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 0 \}\) f. \(\{ \mathbf{x} \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = -1 \}\) g. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = s \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + t \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} \text{ for some } s, t \in \mathbb{R} \}\) h. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix} + s \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + t \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} \text{ for some } s, t \in \mathbb{R} \}\) i. \(\{ \mathbf{x} \in \mathbb{R}^3 : \mathbf
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