- 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
- 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**:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2F3562f10d-07e3-4cc4-8cf1-12dbb5e06a44%2F4q6mwhg_processed.png&w=3840&q=75)
Transcribed Image Text:### 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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