3. Determine if b is a linear combination of a₁, a2, and a3 2 1 2 3 a1 = 1 a2 = 2 , az 1 4 1 0 9 - = 2 b =
3. Determine if b is a linear combination of a₁, a2, and a3 2 1 2 3 a1 = 1 a2 = 2 , az 1 4 1 0 9 - = 2 b =
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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![## Problem Statement
### 3. Determine if \( b \) is a linear combination of \( a_1, a_2, \) and \( a_3 \).
Given vectors:
\[ a_1 = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \quad a_2 = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}, \quad a_3 = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \quad b = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
To determine if vector \( b \) is a linear combination of vectors \( a_1, a_2, \) and \( a_3 \), we need to find scalars \( x, y, \) and \( z \) such that:
\[ x \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + y \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} + z \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
This can be represented as the matrix equation:
\[ x \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + y \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} + z \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
In matrix form, this becomes:
\[ \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 1 \\ 1 & -1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
To solve this system of linear equations, we can use methods such as Gaussian elimination or matrix row reduction. The task is to find if there exists a solution](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d07e113-70c7-44ea-a54a-4dad627b44c8%2F01d2e311-e29e-4919-932c-39c567f5c570%2F6ze8ozc_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
### 3. Determine if \( b \) is a linear combination of \( a_1, a_2, \) and \( a_3 \).
Given vectors:
\[ a_1 = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \quad a_2 = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}, \quad a_3 = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \quad b = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
To determine if vector \( b \) is a linear combination of vectors \( a_1, a_2, \) and \( a_3 \), we need to find scalars \( x, y, \) and \( z \) such that:
\[ x \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + y \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} + z \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
This can be represented as the matrix equation:
\[ x \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} + y \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} + z \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
In matrix form, this becomes:
\[ \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 1 \\ 1 & -1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 0 \end{bmatrix} \]
To solve this system of linear equations, we can use methods such as Gaussian elimination or matrix row reduction. The task is to find if there exists a solution
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