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Show that points on the plane x-2y+4z=0 form a subspace of R3

Show that points on the plane x-2y+4z=0 form a subspace of R3

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Show that points on the plane \( x - 2y + 4z = 0 \) form a subspace of \( \mathbb{R}^3 \). **Hint:** Show closure under addition and scalar multiplication. **Explanation for Educational Context:** To demonstrate that the set of points on the plane \( x - 2y + 4z = 0 \) is a subspace of \( \mathbb{R}^3 \), we need to verify two key properties: 1. **Closure under Addition:** If two vectors \(\mathbf{u} = (x_1, y_1, z_1)\) and \(\mathbf{v} = (x_2, y_2, z_2)\) satisfy the equation, their sum \(\mathbf{u} + \mathbf{v}\) must also satisfy the equation. 2. **Closure under Scalar Multiplication:** If a vector \(\mathbf{u} = (x, y, z)\) satisfies the equation, any scalar multiple \(c\mathbf{u}\) must also satisfy the equation. By proving these properties, we can confirm the set forms a subspace of the vector space \( \mathbb{R}^3 \).
Transcribed Image Text:**Problem Statement:** Show that points on the plane \( x - 2y + 4z = 0 \) form a subspace of \( \mathbb{R}^3 \). **Hint:** Show closure under addition and scalar multiplication. **Explanation for Educational Context:** To demonstrate that the set of points on the plane \( x - 2y + 4z = 0 \) is a subspace of \( \mathbb{R}^3 \), we need to verify two key properties: 1. **Closure under Addition:** If two vectors \(\mathbf{u} = (x_1, y_1, z_1)\) and \(\mathbf{v} = (x_2, y_2, z_2)\) satisfy the equation, their sum \(\mathbf{u} + \mathbf{v}\) must also satisfy the equation. 2. **Closure under Scalar Multiplication:** If a vector \(\mathbf{u} = (x, y, z)\) satisfies the equation, any scalar multiple \(c\mathbf{u}\) must also satisfy the equation. By proving these properties, we can confirm the set forms a subspace of the vector space \( \mathbb{R}^3 \).
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