A basis for the null space of the Matrix 1 2 3 A 2 4 6 is - 1 a. O b. O C. 00 { { { - 3 0 |} 1 2 0 1 2 0 3 0} 1 3 6} 1
A basis for the null space of the Matrix 1 2 3 A 2 4 6 is - 1 a. O b. O C. 00 { { { - 3 0 |} 1 2 0 1 2 0 3 0} 1 3 6} 1
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
![**Finding a Basis for the Null Space of a Matrix**
Given the matrix \( A \):
\[
A = \begin{pmatrix} 1 & -2 & 3 \\ 2 & -4 & 6 \\ 0 & 0 & 1 \end{pmatrix}
\]
We are tasked with identifying a basis for the null space of this matrix.
**Choices:**
- **Option a:**
\[
\begin{Bmatrix}
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\end{Bmatrix}
\]
- **Option b:**
\[
\begin{Bmatrix}
\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix},
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\end{Bmatrix}
\]
- **Option c:**
\[
\begin{Bmatrix}
\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},
\begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}
\end{Bmatrix}
\]
**Detailed Explanation:**
The task is to determine which of these options forms a basis for the null space of matrix \( A \). The null space of a matrix consists of all vectors \( \mathbf{x} \) such that \( A\mathbf{x} = \mathbf{0} \). Finding the basis involves identifying independent vectors that span this subspace of solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c49f70f-02c0-4076-b0a5-564b9a6f1d8d%2Ff306ae6c-2aac-467a-9d06-87660201fb81%2Fssdh4tq_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding a Basis for the Null Space of a Matrix**
Given the matrix \( A \):
\[
A = \begin{pmatrix} 1 & -2 & 3 \\ 2 & -4 & 6 \\ 0 & 0 & 1 \end{pmatrix}
\]
We are tasked with identifying a basis for the null space of this matrix.
**Choices:**
- **Option a:**
\[
\begin{Bmatrix}
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\end{Bmatrix}
\]
- **Option b:**
\[
\begin{Bmatrix}
\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix},
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\end{Bmatrix}
\]
- **Option c:**
\[
\begin{Bmatrix}
\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},
\begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}
\end{Bmatrix}
\]
**Detailed Explanation:**
The task is to determine which of these options forms a basis for the null space of matrix \( A \). The null space of a matrix consists of all vectors \( \mathbf{x} \) such that \( A\mathbf{x} = \mathbf{0} \). Finding the basis involves identifying independent vectors that span this subspace of solutions.
![**A Basis for the Null Space of the Matrix**
Consider the matrix:
\[ A = \begin{pmatrix} 1 & -2 & 3 \\ 2 & -4 & 6 \\ 0 & 0 & 1 \end{pmatrix} \]
Choose the correct basis for the null space of this matrix:
- **Option a:**
\[
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\]
- **Option b:**
\[
\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix},
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\]
- **Option c:**
\[
\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},
\begin{pmatrix} 6 \\ 0 \\ 1 \end{pmatrix}
\]
Each option presents a set of vectors. Select the option that represents a set of linearly independent vectors that form a basis for the null space of the given matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c49f70f-02c0-4076-b0a5-564b9a6f1d8d%2Ff306ae6c-2aac-467a-9d06-87660201fb81%2F3fv0tr9_processed.png&w=3840&q=75)
Transcribed Image Text:**A Basis for the Null Space of the Matrix**
Consider the matrix:
\[ A = \begin{pmatrix} 1 & -2 & 3 \\ 2 & -4 & 6 \\ 0 & 0 & 1 \end{pmatrix} \]
Choose the correct basis for the null space of this matrix:
- **Option a:**
\[
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\]
- **Option b:**
\[
\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix},
\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}
\]
- **Option c:**
\[
\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},
\begin{pmatrix} 6 \\ 0 \\ 1 \end{pmatrix}
\]
Each option presents a set of vectors. Select the option that represents a set of linearly independent vectors that form a basis for the null space of the given matrix.
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