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2. Each set below is a subset of R2. For each set, do the following: • Sketch a picture of the set. • Determine if the set is closed under vector addition. If the answer is yes, state this and justify your answer. If the answer is no, give an example of a pair of vectors V and w that are both in the set but whose sum, v + w, is not in the set. • Determine if the set is closed under scalar multiplication. If the answer is yes, state this and justify your answer. If the answer is no, give an example of a vector v in the set and a scalar c such that cv is not in the set. • Based on your answers above, determine if the set is a subspace of R². The sets: a) The set of vectors (x, y) where y ≥ 0. b) The set including just the origin, (0, 0). c) The set including all points along the y-axis as well as all points along the x-axis. d) The set of vectors (x, y) satifying y = x + 2. e) The set of vectors (x, y) satifying x + 3y = 0.

2. Each set below is a subset of R2. For each set, do the following: • Sketch a picture of the set. • Determine if the set is closed under vector addition. If the answer is yes, state this and justify your answer. If the answer is no, give an example of a pair of vectors V and w that are both in the set but whose sum, v + w, is not in the set. • Determine if the set is closed under scalar multiplication. If the answer is yes, state this and justify your answer. If the answer is no, give an example of a vector v in the set and a scalar c such that cv is not in the set. • Based on your answers above, determine if the set is a subspace of R². The sets: a) The set of vectors (x, y) where y ≥ 0. b) The set including just the origin, (0, 0). c) The set including all points along the y-axis as well as all points along the x-axis. d) The set of vectors (x, y) satifying y = x + 2. e) The set of vectors (x, y) satifying x + 3y = 0.

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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**Exercise: Analyzing Subsets of \(\mathbb{R}^2\)** For each set below, perform the following tasks: - **Sketch** a picture of the set. - **Determine** if the set is closed under vector addition. If yes, state this and justify your answer. If no, provide an example of vectors \(\mathbf{v}\) and \(\mathbf{w}\) within the set whose sum, \(\mathbf{v} + \mathbf{w}\), is not in the set. - **Determine** if the set is closed under scalar multiplication. If yes, state this and justify your answer. If no, provide an example of a vector \(\mathbf{v}\) in the set and a scalar \(c\) such that \(c\mathbf{v}\) is not in the set. - **Conclude** if the set is a subspace of \(\mathbb{R}^2\) based on the above findings. **The Sets:** a) The set of vectors \((x, y)\) where \(y \geq 0\). b) The set including just the origin, \((0, 0)\). c) The set including all points along the y-axis as well as all points along the x-axis. d) The set of vectors \((x, y)\) satisfying \(y = x + 2\). e) The set of vectors \((x, y)\) satisfying \(x + 3y = 0\).
Transcribed Image Text:**Exercise: Analyzing Subsets of \(\mathbb{R}^2\)** For each set below, perform the following tasks: - **Sketch** a picture of the set. - **Determine** if the set is closed under vector addition. If yes, state this and justify your answer. If no, provide an example of vectors \(\mathbf{v}\) and \(\mathbf{w}\) within the set whose sum, \(\mathbf{v} + \mathbf{w}\), is not in the set. - **Determine** if the set is closed under scalar multiplication. If yes, state this and justify your answer. If no, provide an example of a vector \(\mathbf{v}\) in the set and a scalar \(c\) such that \(c\mathbf{v}\) is not in the set. - **Conclude** if the set is a subspace of \(\mathbb{R}^2\) based on the above findings. **The Sets:** a) The set of vectors \((x, y)\) where \(y \geq 0\). b) The set including just the origin, \((0, 0)\). c) The set including all points along the y-axis as well as all points along the x-axis. d) The set of vectors \((x, y)\) satisfying \(y = x + 2\). e) The set of vectors \((x, y)\) satisfying \(x + 3y = 0\).
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