### Linear Algebra Problem: Null Space Determination#### Problem Statement:You are given the following vectors and matrix:\[\mathbf{v_1} = \begin{bmatrix}1 \\6 \\-2 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix}-2 \\6 \\6 \end{bmatrix}, \quad \mathbf{v_3} = \begin{bmatrix}-7 \\7 \\1 \end{bmatrix}, \quad \mathbf{u} = \begin{bmatrix}-7 \\7 \\6 \end{bmatrix}\]Determine if vector **u** is in the null space (Nul) of matrix **A**, where:\[A = \begin{bmatrix}\mathbf{v_1} & \mathbf{v_2} & \mathbf{v_3} \end{bmatrix}\]#### Question:Is **u** in Nul **A**?#### Options:- ⬤ Yes- ⬤ No### Detailed Explanation:1. **Matrix A Construction**: The matrix **A** is constructed by using the vectors $\mathbf{v_1}$, $\mathbf{v_2}$, and $\mathbf{v_3}$ as its columns: \[ A = \begin{bmatrix} 1 & -2 & -7 \\ 6 & 6 & 7 \\ -2 & 6 & 1 \end{bmatrix} \]2. **Null Space**: The null space of a matrix **A** is the set of all vectors $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{0}$. To determine if $\mathbf{u}$ is in Nul **A**, we need to check if there exists a solution to the equation $A\mathbf{c} = \mathbf{u}$, where $\mathbf{u}$ is: \[ \mathbf{u} = \begin{bmatrix} -7 \\ 7 \\ 6 \end{bmatrix} \]3. **Verification Procedure**:

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Answered: - 7 Let v, = 1 6. V3 = -7 and u = 7…

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- 7 Let v, = 1 6. V3 = -7 and u = 7 Determine if u is in Nul A, where A = V1 V2 V3 6 6 7 6 Is u in Nul A? Yes No

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

### Linear Algebra Problem: Null Space Determination

#### Problem Statement:
You are given the following vectors and matrix:

\[
\mathbf{v_1} = \begin{bmatrix}
1 \\
6 \\
-2
\end{bmatrix}, \quad
\mathbf{v_2} = \begin{bmatrix}
-2 \\
6 \\
6
\end{bmatrix}, \quad
\mathbf{v_3} = \begin{bmatrix}
-7 \\
7 \\
1
\end{bmatrix}, \quad
\mathbf{u} = \begin{bmatrix}
-7 \\
7 \\
6
\end{bmatrix}
\]

Determine if vector **u** is in the null space (Nul) of matrix **A**, where:

\[
A = \begin{bmatrix}
\mathbf{v_1} & \mathbf{v_2} & \mathbf{v_3}
\end{bmatrix}
\]

#### Question:
Is **u** in Nul **A**?

#### Options:
- ⬤ Yes
- ⬤ No

### Detailed Explanation:

1. **Matrix A Construction**:
The matrix **A** is constructed by using the vectors $\mathbf{v_1}$, $\mathbf{v_2}$, and $\mathbf{v_3}$ as its columns:

\[
A = \begin{bmatrix}
1 & -2 & -7 \\
6 & 6 & 7 \\
-2 & 6 & 1
\end{bmatrix}
\]

2. **Null Space**:
The null space of a matrix **A** is the set of all vectors $\mathbf{x}$ such that $A\mathbf{x} = \mathbf{0}$. To determine if $\mathbf{u}$ is in Nul **A**, we need to check if there exists a solution to the equation $A\mathbf{c} = \mathbf{u}$, where $\mathbf{u}$ is:

\[
\mathbf{u} = \begin{bmatrix}
-7 \\
7 \\
6
\end{bmatrix}
\]

3. **Verification Procedure**:

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