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Let S = {v1, V2, V3} C R³ be a set of linearly independent vectors. Show that S is a basis for R3.

Let S = {v1, V2, V3} C R³ be a set of linearly independent vectors. Show that S is a basis for R3.

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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I have no idea how to work on this problem I am not doing bad in class but this one is complicated for an online course that I am self teaching myself if you could help please 

**Problem Statement:** Let \( S = \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \subseteq \mathbb{R}^3 \) be a set of linearly independent vectors. Show that \( S \) is a basis for \( \mathbb{R}^3 \). **Explanation:** This statement posits a mathematical problem concerning vector spaces. Here, the set \( S \) comprises three vectors, \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \), which are elements of the three-dimensional real coordinate space \( \mathbb{R}^3 \). To demonstrate that \( S \) constitutes a basis for \( \mathbb{R}^3 \), we must show two properties: 1. **Linearly Independent:** This condition is already given in the problem statement. 2. **Spanning Set:** The vectors in \( S \) must span \( \mathbb{R}^3 \), meaning every vector in \( \mathbb{R}^3 \) can be expressed as a linear combination of \( \mathbf{v}_1, \mathbf{v}_2, \) and \( \mathbf{v}_3 \). Given \( S \) is a basis if it simultaneously satisfies these conditions.
Transcribed Image Text:**Problem Statement:** Let \( S = \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \subseteq \mathbb{R}^3 \) be a set of linearly independent vectors. Show that \( S \) is a basis for \( \mathbb{R}^3 \). **Explanation:** This statement posits a mathematical problem concerning vector spaces. Here, the set \( S \) comprises three vectors, \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \), which are elements of the three-dimensional real coordinate space \( \mathbb{R}^3 \). To demonstrate that \( S \) constitutes a basis for \( \mathbb{R}^3 \), we must show two properties: 1. **Linearly Independent:** This condition is already given in the problem statement. 2. **Spanning Set:** The vectors in \( S \) must span \( \mathbb{R}^3 \), meaning every vector in \( \mathbb{R}^3 \) can be expressed as a linear combination of \( \mathbf{v}_1, \mathbf{v}_2, \) and \( \mathbf{v}_3 \). Given \( S \) is a basis if it simultaneously satisfies these conditions.
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