Construct a matrix A for which the nullspace is the set of all linear combinations of 6. 8 1 and 1 3
Construct a matrix A for which the nullspace is the set of all linear combinations of 6. 8 1 and 1 3
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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![**Task: Construct a Matrix with a Given Nullspace**
**Objective:** Construct a matrix \( A \) for which the nullspace is the set of all linear combinations of the vectors:
\[
\begin{bmatrix}
6 \\
1 \\
0 \\
3
\end{bmatrix}
\]
and
\[
\begin{bmatrix}
8 \\
0 \\
1 \\
2
\end{bmatrix}
\]
**Explanation:**
You are tasked with finding a matrix \( A \) such that any linear combination of the given vectors results in a vector in the nullspace of \( A \). The nullspace of \( A \) is the set of all vectors \( \mathbf{v} \) such that \( A\mathbf{v} = \mathbf{0} \).
**Steps to Solve:**
1. **Identify the Nullspace Vectors:** The vectors given are
\[
\mathbf{v_1} = \begin{bmatrix} 6 \\ 1 \\ 0 \\ 3 \end{bmatrix}, \quad
\mathbf{v_2} = \begin{bmatrix} 8 \\ 0 \\ 1 \\ 2 \end{bmatrix}
\]
2. **Form the Nullspace Matrix:** Create a matrix \( N \) where these vectors are the columns:
\[
N = \begin{bmatrix}
6 & 8 \\
1 & 0 \\
0 & 1 \\
3 & 2
\end{bmatrix}
\]
3. **Determine \( A \):** To find matrix \( A \), ensure that \( N \) is in the nullspace, i.e., \( AN = 0 \). This typically involves finding a matrix \( A \) such that each column of \( N \) is mapped to the zero vector.
Using the nullspace properties, you may use Gaussian elimination or other methods to derive \( A \).
In this context, your task is to derive such a matrix \( A \), ensuring your logic matches the conditions of the problem, i.e., the nullspace is spanned by \(\mathbf{v_1}\) and \(\mathbf{v_2}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7297a90f-0927-4369-b782-4368eb3beb88%2F38518027-d32a-44cf-afc9-002646f8f784%2Fn25d64x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Construct a Matrix with a Given Nullspace**
**Objective:** Construct a matrix \( A \) for which the nullspace is the set of all linear combinations of the vectors:
\[
\begin{bmatrix}
6 \\
1 \\
0 \\
3
\end{bmatrix}
\]
and
\[
\begin{bmatrix}
8 \\
0 \\
1 \\
2
\end{bmatrix}
\]
**Explanation:**
You are tasked with finding a matrix \( A \) such that any linear combination of the given vectors results in a vector in the nullspace of \( A \). The nullspace of \( A \) is the set of all vectors \( \mathbf{v} \) such that \( A\mathbf{v} = \mathbf{0} \).
**Steps to Solve:**
1. **Identify the Nullspace Vectors:** The vectors given are
\[
\mathbf{v_1} = \begin{bmatrix} 6 \\ 1 \\ 0 \\ 3 \end{bmatrix}, \quad
\mathbf{v_2} = \begin{bmatrix} 8 \\ 0 \\ 1 \\ 2 \end{bmatrix}
\]
2. **Form the Nullspace Matrix:** Create a matrix \( N \) where these vectors are the columns:
\[
N = \begin{bmatrix}
6 & 8 \\
1 & 0 \\
0 & 1 \\
3 & 2
\end{bmatrix}
\]
3. **Determine \( A \):** To find matrix \( A \), ensure that \( N \) is in the nullspace, i.e., \( AN = 0 \). This typically involves finding a matrix \( A \) such that each column of \( N \) is mapped to the zero vector.
Using the nullspace properties, you may use Gaussian elimination or other methods to derive \( A \).
In this context, your task is to derive such a matrix \( A \), ensuring your logic matches the conditions of the problem, i.e., the nullspace is spanned by \(\mathbf{v_1}\) and \(\mathbf{v_2}\).
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