Determine whether b is in the column space of A, and if so, express b as a linear combination of the column vectors of A. A = b = -10 bis in the column space of A and b = + 6 -10 bis in the column space of A and b = + O bis not in the column space of A bis in the column space of A and b = bis in the column space of A and b = O O
Determine whether b is in the column space of A, and if so, express b as a linear combination of the column vectors of A. A = b = -10 bis in the column space of A and b = + 6 -10 bis in the column space of A and b = + O bis not in the column space of A bis in the column space of A and b = bis in the column space of A and b = O O
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
Determine whether vector \( \mathbf{b} \) is in the column space of matrix \( \mathbf{A} \), and if so, express \( \mathbf{b} \) as a linear combination of the column vectors of \( \mathbf{A} \).
Given:
\[
\mathbf{A} = \begin{bmatrix} 3 & 5 \\ 6 & -10 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} -2 \\ 16 \end{bmatrix}
\]
## Options
- **Option 1:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 3 \\ 6 \end{bmatrix} + \begin{bmatrix} 5 \\ -10 \end{bmatrix} \)
- **Option 2:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 3 \\ 5 \end{bmatrix} + \begin{bmatrix} 6 \\ -10 \end{bmatrix} \) **(Selected Answer)**
- **Option 3:** \( \mathbf{b} \) is not in the column space of \( \mathbf{A} \)
- **Option 4:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 6 \\ -10 \end{bmatrix} - \begin{bmatrix} 5 \\ -10 \end{bmatrix} \)
- **Option 5:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 3 \\ 5 \end{bmatrix} - \begin{bmatrix} 6 \\ -10 \end{bmatrix} \)
## Explanation
The task requires checking if \( \mathbf{b} \) can be expressed as a linear combination of the columns of \( \mathbf{A} \). The columns of \( \mathbf{A} \) are \( \begin](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b53ebc-804c-49f3-b917-c4e1c4975b52%2F27faaf89-1aac-4db8-b41a-3e45ed16456e%2Fep9scbg_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
Determine whether vector \( \mathbf{b} \) is in the column space of matrix \( \mathbf{A} \), and if so, express \( \mathbf{b} \) as a linear combination of the column vectors of \( \mathbf{A} \).
Given:
\[
\mathbf{A} = \begin{bmatrix} 3 & 5 \\ 6 & -10 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} -2 \\ 16 \end{bmatrix}
\]
## Options
- **Option 1:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 3 \\ 6 \end{bmatrix} + \begin{bmatrix} 5 \\ -10 \end{bmatrix} \)
- **Option 2:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 3 \\ 5 \end{bmatrix} + \begin{bmatrix} 6 \\ -10 \end{bmatrix} \) **(Selected Answer)**
- **Option 3:** \( \mathbf{b} \) is not in the column space of \( \mathbf{A} \)
- **Option 4:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 6 \\ -10 \end{bmatrix} - \begin{bmatrix} 5 \\ -10 \end{bmatrix} \)
- **Option 5:** \( \mathbf{b} \) is in the column space of \( \mathbf{A} \) and \( \mathbf{b} = \begin{bmatrix} 3 \\ 5 \end{bmatrix} - \begin{bmatrix} 6 \\ -10 \end{bmatrix} \)
## Explanation
The task requires checking if \( \mathbf{b} \) can be expressed as a linear combination of the columns of \( \mathbf{A} \). The columns of \( \mathbf{A} \) are \( \begin
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