Write v as a linear combination of u and w, if possible, where u = (3, 1) and w=(3,-2). (Enter your answer in terms of u and w. If not possible, enter IMPOSSIBLE.) v = (6, -1) (u+w) V = Need Help? Read It Watch It Viewing Saved Work Revert to Last Response
Write v as a linear combination of u and w, if possible, where u = (3, 1) and w=(3,-2). (Enter your answer in terms of u and w. If not possible, enter IMPOSSIBLE.) v = (6, -1) (u+w) V = Need Help? Read It Watch It Viewing Saved Work Revert to Last Response
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
![**Title: Linear Combination of Vectors**
**Linear Combination Problem Statement:**
Given vectors \( \mathbf{u} \), \( \mathbf{w} \), and \( \mathbf{v} \), the objective is to express \( \mathbf{v} \) as a linear combination of \( \mathbf{u} \) and \( \mathbf{w} \), if possible.
**Given Vectors:**
- \( \mathbf{u} = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \)
- \( \mathbf{w} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} \)
- \( \mathbf{v} = \begin{pmatrix} 6 \\ -1 \end{pmatrix} \)
**Task:**
Write \( \mathbf{v} \) as a linear combination of \( \mathbf{u} \) and \( \mathbf{w} \).
**Equation:**
\[ \mathbf{v} = c_1 \mathbf{u} + c_2 \mathbf{w} \]
Where:
- \( \mathbf{v} \): the target vector.
- \( \mathbf{u} \) and \( \mathbf{w} \): the basis vectors.
- \( c_1 \) and \( c_2 \): coefficients to determine.
**Solution Format:**
- Express the solution in terms of the given vectors \( \mathbf{u} \) and \( \mathbf{w} \).
**Instructions:**
1. Use vector addition to find the coefficients \( c_1 \) and \( c_2 \).
2. Verify if the linear combination is possible. If not, state "IMPOSSIBLE".
For assistance:
- Click "Read It" for a detailed textual explanation.
- Click "Watch It" for a visual tutorial.
**Example:**
The problem requires finding constants \( c_1 \) and \( c_2 \) such that:
\[ \mathbf{v} = c_1 \mathbf{u} + c_2 \mathbf{w} \]
Where:
\[ \mathbf{v} = \begin{pmatrix} 6 \\ -1 \end{pmatrix}, \]
\[ \mathbf{u} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}, \]
\[ \mathbf](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e1fa912-5e0c-4d6e-bcb6-bc3942e62084%2Fb289b75c-46ce-4bf6-8a64-0d7e51534454%2F36l9df7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Linear Combination of Vectors**
**Linear Combination Problem Statement:**
Given vectors \( \mathbf{u} \), \( \mathbf{w} \), and \( \mathbf{v} \), the objective is to express \( \mathbf{v} \) as a linear combination of \( \mathbf{u} \) and \( \mathbf{w} \), if possible.
**Given Vectors:**
- \( \mathbf{u} = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \)
- \( \mathbf{w} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} \)
- \( \mathbf{v} = \begin{pmatrix} 6 \\ -1 \end{pmatrix} \)
**Task:**
Write \( \mathbf{v} \) as a linear combination of \( \mathbf{u} \) and \( \mathbf{w} \).
**Equation:**
\[ \mathbf{v} = c_1 \mathbf{u} + c_2 \mathbf{w} \]
Where:
- \( \mathbf{v} \): the target vector.
- \( \mathbf{u} \) and \( \mathbf{w} \): the basis vectors.
- \( c_1 \) and \( c_2 \): coefficients to determine.
**Solution Format:**
- Express the solution in terms of the given vectors \( \mathbf{u} \) and \( \mathbf{w} \).
**Instructions:**
1. Use vector addition to find the coefficients \( c_1 \) and \( c_2 \).
2. Verify if the linear combination is possible. If not, state "IMPOSSIBLE".
For assistance:
- Click "Read It" for a detailed textual explanation.
- Click "Watch It" for a visual tutorial.
**Example:**
The problem requires finding constants \( c_1 \) and \( c_2 \) such that:
\[ \mathbf{v} = c_1 \mathbf{u} + c_2 \mathbf{w} \]
Where:
\[ \mathbf{v} = \begin{pmatrix} 6 \\ -1 \end{pmatrix}, \]
\[ \mathbf{u} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}, \]
\[ \mathbf
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