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1) Prove both versions of the triangle inequality: for all a, b = R", we have ||a + b|| ≤ ||a|| + || b || and ||a|| – ||b|| ≤ ||a – b||.

1) Prove both versions of the triangle inequality: for all a, b = R", we have ||a + b|| ≤ ||a|| + || b || and ||a|| – ||b|| ≤ ||a – b||.

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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**Triangle Inequality Proof** **Objective:** Prove both versions of the triangle inequality. For all vectors **a**, **b** in \(\mathbb{R}^n\), we have the following inequalities: 1. \(\| \mathbf{a} + \mathbf{b} \| \leq \| \mathbf{a} \| + \| \mathbf{b} \|\) 2. \(\left| \| \mathbf{a} \| - \| \mathbf{b} \| \right| \leq \| \mathbf{a} - \mathbf{b} \|\) These inequalities illustrate fundamental properties of vector addition and subtraction in n-dimensional real space. The first inequality shows that the length of the sum of two vectors is no greater than the sum of their lengths. The second ensures the absolute difference in their lengths is no greater than the length of their difference. A deeper understanding of these inequalities can be gleaned from exploring their implications in various mathematical contexts such as geometry and functional analysis.
Transcribed Image Text:**Triangle Inequality Proof** **Objective:** Prove both versions of the triangle inequality. For all vectors **a**, **b** in \(\mathbb{R}^n\), we have the following inequalities: 1. \(\| \mathbf{a} + \mathbf{b} \| \leq \| \mathbf{a} \| + \| \mathbf{b} \|\) 2. \(\left| \| \mathbf{a} \| - \| \mathbf{b} \| \right| \leq \| \mathbf{a} - \mathbf{b} \|\) These inequalities illustrate fundamental properties of vector addition and subtraction in n-dimensional real space. The first inequality shows that the length of the sum of two vectors is no greater than the sum of their lengths. The second ensures the absolute difference in their lengths is no greater than the length of their difference. A deeper understanding of these inequalities can be gleaned from exploring their implications in various mathematical contexts such as geometry and functional analysis.
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