### Vectors in the Spanned Space#### Problem StatementLet $ V = \{ \langle 2, -4, -5 \rangle , \langle -4, 2, -5 \rangle \} $. Which of the following are vectors in the space spanned by $ V $?- $ \langle -47, -5, -52 \rangle $- $ \langle -22, 56, 85 \rangle $- $ \langle 20, -3, 21 \rangle $- $ \langle 2, -4, -5 \rangle $- $ \langle 37, 24, 17 \rangle $### ExplanationHere, you are provided with a set $ V $ consisting of two vectors: $ \langle 2, -4, -5 \rangle $ and $ \langle -4, 2, -5 \rangle $. You need to determine which of the given vectors can be expressed as a linear combination of the vectors in $ V $.A vector $ \mathbf{w} $ is said to be in the space spanned by $ V $ if there exist scalars $ a $ and $ b $ such that:\[ a \langle 2, -4, -5 \rangle + b \langle -4, 2, -5 \rangle = \mathbf{w} \]This problem is an example of determining linear dependence in linear algebra.

Answered: Image /qna-images/answer/c1f1faca-b08c-4297-b919-8ec22632c5d0.jpg

Let V = {(2, – 4, – 5), ( – 4, 2, – 5)}. Which of the following are vectors in the space spanned by V? n(- 47, – 5, – 52) n(- 22, 56, 85) п (20, — 3, 21) п (2, — 4, — 5) - n(37, 24, 17)

Let V = {(2, – 4, – 5), ( – 4, 2, – 5)}. Which of the following are vectors in the space spanned by V? n(- 47, – 5, – 52) n(- 22, 56, 85) п (20, — 3, 21) п (2, — 4, — 5) - n(37, 24, 17)

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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Concept explainers

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

### Vectors in the Spanned Space

#### Problem Statement
Let $ V = \{ \langle 2, -4, -5 \rangle , \langle -4, 2, -5 \rangle \} $. Which of the following are vectors in the space spanned by $ V $?

- $ \langle -47, -5, -52 \rangle $
- $ \langle -22, 56, 85 \rangle $
- $ \langle 20, -3, 21 \rangle $
- $ \langle 2, -4, -5 \rangle $
- $ \langle 37, 24, 17 \rangle $

### Explanation
Here, you are provided with a set $ V $ consisting of two vectors: $ \langle 2, -4, -5 \rangle $ and $ \langle -4, 2, -5 \rangle $. You need to determine which of the given vectors can be expressed as a linear combination of the vectors in $ V $.

A vector $ \mathbf{w} $ is said to be in the space spanned by $ V $ if there exist scalars $ a $ and $ b $ such that:
\[ a \langle 2, -4, -5 \rangle + b \langle -4, 2, -5 \rangle = \mathbf{w} \]

This problem is an example of determining linear dependence in linear algebra.

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