the first 2 are wrong, please help me find the right answer ## Problem StatementDetermine 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} $## ExplanationThe 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

Name: the first 2 are wrong, please help me find the right answer ## Problem
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Description: the first 2 are wrong, please help me find the right answer ## Problem StatementDetermine 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 vec

Given a matrix A=356-10 and a vector b = -216. Express b as a linear combination of the column…

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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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the first 2 are wrong, please help me find the right answer

## 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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