Suppose for vectors u, v and w that – 3u + 5v + 2w = 0. What does this say about the set {u, v, w}? Answer: There is not enough information to determine whether the set is linearly independent or linearly dependent. O It is linearly independent It is linearly dependent

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Linear Independence of Vectors

**Problem Statement:**

Suppose for vectors \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \) that 

\[ -3\vec{u} + 5\vec{v} + 2\vec{w} = \vec{0}. \]

What does this say about the set \( \{ \vec{u}, \vec{v}, \vec{w} \} \)?

**Answer:**
- There is not enough information to determine whether the set is linearly independent or linearly dependent.
- It is linearly independent.
- It is linearly dependent.

**Explanation:**

The equation provided is:

\[ -3\vec{u} + 5\vec{v} + 2\vec{w} = \vec{0}. \]

This represents a linear combination of vectors \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \) resulting in the zero vector. The coefficients \(-3\), \(5\), and \(2\) are non-zero, indicating that a non-trivial linear combination of the vectors results in the zero vector. Therefore, the vectors \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \) are linearly dependent.

_Selecting the correct answer:_

- It is linearly dependent.
Transcribed Image Text:### Linear Independence of Vectors **Problem Statement:** Suppose for vectors \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \) that \[ -3\vec{u} + 5\vec{v} + 2\vec{w} = \vec{0}. \] What does this say about the set \( \{ \vec{u}, \vec{v}, \vec{w} \} \)? **Answer:** - There is not enough information to determine whether the set is linearly independent or linearly dependent. - It is linearly independent. - It is linearly dependent. **Explanation:** The equation provided is: \[ -3\vec{u} + 5\vec{v} + 2\vec{w} = \vec{0}. \] This represents a linear combination of vectors \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \) resulting in the zero vector. The coefficients \(-3\), \(5\), and \(2\) are non-zero, indicating that a non-trivial linear combination of the vectors results in the zero vector. Therefore, the vectors \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \) are linearly dependent. _Selecting the correct answer:_ - It is linearly dependent.
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