### 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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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

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)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

### 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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