10 Let a₁ = , a₂ = a35 , and b = Determine whether b can be written as a linear combination of a₁, a2, and a3. In other words, determine whether weights xX₁, X2, and x3 exist, such that x₁ a₁ + x2 a2 + x3 a3 = b. Determine the weights X₁, X2, and x3 if possible. Ox₁=-3, x₂ = 0, x3 = 1 No solution OX₁-2, X₂=-1, X3 = -6 0x₁=2₁x₂=1, X3-0 A
10 Let a₁ = , a₂ = a35 , and b = Determine whether b can be written as a linear combination of a₁, a2, and a3. In other words, determine whether weights xX₁, X2, and x3 exist, such that x₁ a₁ + x2 a2 + x3 a3 = b. Determine the weights X₁, X2, and x3 if possible. Ox₁=-3, x₂ = 0, x3 = 1 No solution OX₁-2, X₂=-1, X3 = -6 0x₁=2₁x₂=1, X3-0 A
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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### Linear Combinations and Vector Spaces
**Problem Statement:**
Given the vectors and the target vector:
\[
a_1 = \begin{bmatrix} 1 \\ 2 \\ -3 \end{bmatrix}, \quad
a_2 = \begin{bmatrix} -3 \\ -4 \\ 1 \end{bmatrix}, \quad
a_3 = \begin{bmatrix} 2 \\ 1 \\ 6 \end{bmatrix}, \quad
b = \begin{bmatrix} -1 \\ -6 \\ 1 \end{bmatrix}
\]
Determine whether the vector \( b \) can be written as a linear combination of \( a_1, a_2, \) and \( a_3 \). In other words, find weights \( x_1, x_2, \) and \( x_3 \) such that:
\[
x_1 a_1 + x_2 a_2 + x_3 a_3 = b
\]
**Options:**
- \( x_1 = -3, x_2 = 0, x_3 = 1 \)
- No solution
- \( x_1 = -2, x_2 = -1, x_3 = -6 \)
- \( x_1 = 2, x_2 = 1, x_3 = 0 \)
### Solution Approach:
To determine whether \( b \) is a linear combination of \( a_1, a_2, \) and \( a_3 \), we need to solve the following system of linear equations:
\[
x_1 \begin{bmatrix} 1 \\ 2 \\ -3 \end{bmatrix} +
x_2 \begin{bmatrix} -3 \\ -4 \\ 1 \end{bmatrix} +
x_3 \begin{bmatrix} 2 \\ 1 \\ 6 \end{bmatrix} = \begin{bmatrix} -1 \\ -6 \\ 1 \end{bmatrix}
\]
Breaking this down into individual equations, we get:
1. \( x_1 - 3x_2 + 2x_3 = -1 \)
2. \( 2x_1 - 4x_2 + x_3 = -6 \)
3. \( -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F157e66f5-e794-4af2-988e-1885b818a80a%2F28881b3b-67ec-4732-8613-80a9183acff0%2Fshdkhbv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Linear Combinations and Vector Spaces
**Problem Statement:**
Given the vectors and the target vector:
\[
a_1 = \begin{bmatrix} 1 \\ 2 \\ -3 \end{bmatrix}, \quad
a_2 = \begin{bmatrix} -3 \\ -4 \\ 1 \end{bmatrix}, \quad
a_3 = \begin{bmatrix} 2 \\ 1 \\ 6 \end{bmatrix}, \quad
b = \begin{bmatrix} -1 \\ -6 \\ 1 \end{bmatrix}
\]
Determine whether the vector \( b \) can be written as a linear combination of \( a_1, a_2, \) and \( a_3 \). In other words, find weights \( x_1, x_2, \) and \( x_3 \) such that:
\[
x_1 a_1 + x_2 a_2 + x_3 a_3 = b
\]
**Options:**
- \( x_1 = -3, x_2 = 0, x_3 = 1 \)
- No solution
- \( x_1 = -2, x_2 = -1, x_3 = -6 \)
- \( x_1 = 2, x_2 = 1, x_3 = 0 \)
### Solution Approach:
To determine whether \( b \) is a linear combination of \( a_1, a_2, \) and \( a_3 \), we need to solve the following system of linear equations:
\[
x_1 \begin{bmatrix} 1 \\ 2 \\ -3 \end{bmatrix} +
x_2 \begin{bmatrix} -3 \\ -4 \\ 1 \end{bmatrix} +
x_3 \begin{bmatrix} 2 \\ 1 \\ 6 \end{bmatrix} = \begin{bmatrix} -1 \\ -6 \\ 1 \end{bmatrix}
\]
Breaking this down into individual equations, we get:
1. \( x_1 - 3x_2 + 2x_3 = -1 \)
2. \( 2x_1 - 4x_2 + x_3 = -6 \)
3. \( -
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