Can vector w = 3 be written as a linear combination of v₁ = 4 4 vector w belong to span{v₁,v₂}? V₂ || 3? In other words does 2
Can vector w = 3 be written as a linear combination of v₁ = 4 4 vector w belong to span{v₁,v₂}? V₂ || 3? In other words does 2
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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Question
![**Understanding Linear Combinations and Vector Span**
This section of the educational website will delve into the concepts of linear combinations and vector spans.
### Problem Statement:
**Question:**
Can vector \( w \) be written as a linear combination of \( \mathbf{v_1} \) and \( \mathbf{v_2} \)?
Given vectors:
\[ w = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \]
\[ \mathbf{v_1} = \begin{bmatrix} 3 \\ 1 \\ 4 \end{bmatrix}, \mathbf{v_2} = \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix} \]
In other words, does vector \( w \) belong to \(\text{span}\{\mathbf{v_1}, \mathbf{v_2}\}\)?
### Explanation:
To determine if \( w \) can be expressed as a linear combination of \( \mathbf{v_1} \) and \( \mathbf{v_2} \), we need to find if there exist scalars \( a \) and \( b \) such that:
\[ w = a\mathbf{v_1} + b\mathbf{v_2} \]
This translates to the following linear system of equations when equated component-wise:
\[ \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} = a \begin{bmatrix} 3 \\ 1 \\ 4 \end{bmatrix} + b \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix} \]
\[ \Rightarrow 2 = 3a + b \]
\[ 3 = a + 3b \]
\[ 4 = 4a - 2b \]
We need to solve this system of equations to find the values of \( a \) and \( b \). If the system has a solution, then \( w \) lies in the span of \( \mathbf{v_1} \) and \( \mathbf{v_2} \).
### Detailed Steps:
1. Set up the augmented matrix for the system:
\[
\begin{pmatrix}
3 & 1 & | & 2 \\
1 & 3 & | & 3 \\
4 & -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff869c4ac-8e58-4431-846d-0202389eaebe%2Ffdbb4a66-ecbd-4803-80cc-9a7870f95ff1%2Fcmwfxzf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Linear Combinations and Vector Span**
This section of the educational website will delve into the concepts of linear combinations and vector spans.
### Problem Statement:
**Question:**
Can vector \( w \) be written as a linear combination of \( \mathbf{v_1} \) and \( \mathbf{v_2} \)?
Given vectors:
\[ w = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \]
\[ \mathbf{v_1} = \begin{bmatrix} 3 \\ 1 \\ 4 \end{bmatrix}, \mathbf{v_2} = \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix} \]
In other words, does vector \( w \) belong to \(\text{span}\{\mathbf{v_1}, \mathbf{v_2}\}\)?
### Explanation:
To determine if \( w \) can be expressed as a linear combination of \( \mathbf{v_1} \) and \( \mathbf{v_2} \), we need to find if there exist scalars \( a \) and \( b \) such that:
\[ w = a\mathbf{v_1} + b\mathbf{v_2} \]
This translates to the following linear system of equations when equated component-wise:
\[ \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} = a \begin{bmatrix} 3 \\ 1 \\ 4 \end{bmatrix} + b \begin{bmatrix} 1 \\ 3 \\ -2 \end{bmatrix} \]
\[ \Rightarrow 2 = 3a + b \]
\[ 3 = a + 3b \]
\[ 4 = 4a - 2b \]
We need to solve this system of equations to find the values of \( a \) and \( b \). If the system has a solution, then \( w \) lies in the span of \( \mathbf{v_1} \) and \( \mathbf{v_2} \).
### Detailed Steps:
1. Set up the augmented matrix for the system:
\[
\begin{pmatrix}
3 & 1 & | & 2 \\
1 & 3 & | & 3 \\
4 & -
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