Solve the problem. -18 H Let a₁ a₂ , and b 17 Determine whether b can be written as a linear combination of a1 and a2. In other words, determine whether weights x₁ and x₂ exist, such that x₁ a₁ + x₂ a₂ = b. Determine the weights x₁ and x₂ if possible. O x₁ = -3, x₂ = 4 O No solution O x1 = -2, x₂ = 3 O x1 = -3, x₂ = 5
Solve the problem. -18 H Let a₁ a₂ , and b 17 Determine whether b can be written as a linear combination of a1 and a2. In other words, determine whether weights x₁ and x₂ exist, such that x₁ a₁ + x₂ a₂ = b. Determine the weights x₁ and x₂ if possible. O x₁ = -3, x₂ = 4 O No solution O x1 = -2, x₂ = 3 O x1 = -3, x₂ = 5
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
![**Problem Statement:**
**Objective:**
Determine if the vector \( b \) can be expressed as a linear combination of the vectors \( a_1 \) and \( a_2 \). In other words, find whether there exist scalars \( x_1 \) and \( x_2 \) such that:
\[ x_1 a_1 + x_2 a_2 = b \]
If possible, determine the weights \( x_1 \) and \( x_2 \).
**Given:**
\[ a_1 = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \quad a_2 = \begin{bmatrix} -3 \\ 2 \end{bmatrix}, \quad b = \begin{bmatrix} -18 \\ 17 \end{bmatrix} \]
**Solution:**
1. Choose the correct option for \( x_1 \) and \( x_2 \) from the following choices:
- \( x_1 = -3, x_2 = 5 \)
- \( x_1 = -2, x_2 = 3 \)
- \( x_1 = -3, x_2 = 4 \)
- No solution
**Explanation:**
To determine if \( b \) can be expressed as a linear combination of \( a_1 \) and \( a_2 \), we need to solve the following system of linear equations derived from the expression \( x_1 a_1 + x_2 a_2 = b \):
\[ x_1 \begin{bmatrix} 2 \\ 3 \end{bmatrix} + x_2 \begin{bmatrix} -3 \\ 2 \end{bmatrix} = \begin{bmatrix} -18 \\ 17 \end{bmatrix} \]
This results in the system:
\[
2x_1 - 3x_2 = -18 \\
3x_1 + 2x_2 = 17
\]
Solving this system will give us the weights \( x_1 \) and \( x_2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F157e66f5-e794-4af2-988e-1885b818a80a%2Fcdc692ea-91c7-4a54-bbca-a6a77af386c9%2Fxzxspxe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
**Objective:**
Determine if the vector \( b \) can be expressed as a linear combination of the vectors \( a_1 \) and \( a_2 \). In other words, find whether there exist scalars \( x_1 \) and \( x_2 \) such that:
\[ x_1 a_1 + x_2 a_2 = b \]
If possible, determine the weights \( x_1 \) and \( x_2 \).
**Given:**
\[ a_1 = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \quad a_2 = \begin{bmatrix} -3 \\ 2 \end{bmatrix}, \quad b = \begin{bmatrix} -18 \\ 17 \end{bmatrix} \]
**Solution:**
1. Choose the correct option for \( x_1 \) and \( x_2 \) from the following choices:
- \( x_1 = -3, x_2 = 5 \)
- \( x_1 = -2, x_2 = 3 \)
- \( x_1 = -3, x_2 = 4 \)
- No solution
**Explanation:**
To determine if \( b \) can be expressed as a linear combination of \( a_1 \) and \( a_2 \), we need to solve the following system of linear equations derived from the expression \( x_1 a_1 + x_2 a_2 = b \):
\[ x_1 \begin{bmatrix} 2 \\ 3 \end{bmatrix} + x_2 \begin{bmatrix} -3 \\ 2 \end{bmatrix} = \begin{bmatrix} -18 \\ 17 \end{bmatrix} \]
This results in the system:
\[
2x_1 - 3x_2 = -18 \\
3x_1 + 2x_2 = 17
\]
Solving this system will give us the weights \( x_1 \) and \( x_2 \).
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