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Prove: For nonzero vectors ū,vER, if (dot product) ū-v=0, then ū & v are linearly independent. Hint: For nonzero vector w-w+0.

Prove: For nonzero vectors ū,vER, if (dot product) ū-v=0, then ū & v are linearly independent. Hint: For nonzero vector w-w+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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**Title: Proof of Linear Independence for Orthogonal Vectors in \( \mathbb{R}^3 \)** **Objective:** To prove that for nonzero vectors \( \mathbf{\tilde{u}}, \mathbf{\tilde{v}} \in \mathbb{R}^3 \), if their dot product \( \mathbf{\tilde{u}} \cdot \mathbf{\tilde{v}} = 0 \), then the vectors \( \mathbf{\tilde{u}} \) and \( \mathbf{\tilde{v}} \) are linearly independent. **Hint Provided:** For a nonzero vector \( \mathbf{\tilde{w}} \), the dot product \( \mathbf{\tilde{w}} \cdot \mathbf{\tilde{w}} \neq 0 \). --- **Explanation:** This theorem addresses the relationship between the dot product of two vectors and their linear independence. Specifically, it states that if two vectors in three-dimensional space are orthogonal (i.e., their dot product is zero), they are assured to be linearly independent, provided neither is the zero vector. **Important Concept:** - **Orthogonality:** Two vectors are orthogonal if their dot product is zero. In \( \mathbb{R}^3 \), this means they form a right angle with respect to each other. - **Linear Independence:** Two vectors are linearly independent if no scalar multiple of one vector can be expressed as another vector unless the scalar is zero. In simpler terms, neither vector lies along the line spanned by the other. By understanding the nature of dot products and linear independence, students will be able to prove the stated theorem using these fundamental concepts in vector algebra. The provided hint emphasizes the non-zero nature of the dot product of a vector with itself, which is a stepping stone to understanding why orthogonal vectors in \( \mathbb{R}^3 \) lead to linear independence.
Transcribed Image Text:**Title: Proof of Linear Independence for Orthogonal Vectors in \( \mathbb{R}^3 \)** **Objective:** To prove that for nonzero vectors \( \mathbf{\tilde{u}}, \mathbf{\tilde{v}} \in \mathbb{R}^3 \), if their dot product \( \mathbf{\tilde{u}} \cdot \mathbf{\tilde{v}} = 0 \), then the vectors \( \mathbf{\tilde{u}} \) and \( \mathbf{\tilde{v}} \) are linearly independent. **Hint Provided:** For a nonzero vector \( \mathbf{\tilde{w}} \), the dot product \( \mathbf{\tilde{w}} \cdot \mathbf{\tilde{w}} \neq 0 \). --- **Explanation:** This theorem addresses the relationship between the dot product of two vectors and their linear independence. Specifically, it states that if two vectors in three-dimensional space are orthogonal (i.e., their dot product is zero), they are assured to be linearly independent, provided neither is the zero vector. **Important Concept:** - **Orthogonality:** Two vectors are orthogonal if their dot product is zero. In \( \mathbb{R}^3 \), this means they form a right angle with respect to each other. - **Linear Independence:** Two vectors are linearly independent if no scalar multiple of one vector can be expressed as another vector unless the scalar is zero. In simpler terms, neither vector lies along the line spanned by the other. By understanding the nature of dot products and linear independence, students will be able to prove the stated theorem using these fundamental concepts in vector algebra. The provided hint emphasizes the non-zero nature of the dot product of a vector with itself, which is a stepping stone to understanding why orthogonal vectors in \( \mathbb{R}^3 \) lead to linear independence.
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