Use the fact that matrices A and B are row-equivalent. 1 2 1 2 A = 5 1 7 2 2 -2 5 11 4 -4 10 1 0 3 0 -4 0 1 -1 0 2 B = 0 0 0 0 0 1 -2 0 0 (a) Find the rank and nullity of A. rank nullity (b) Find a basis for the nullspace of A. (c) Find a basis for the row space of A.
Use the fact that matrices A and B are row-equivalent. 1 2 1 2 A = 5 1 7 2 2 -2 5 11 4 -4 10 1 0 3 0 -4 0 1 -1 0 2 B = 0 0 0 0 0 1 -2 0 0 (a) Find the rank and nullity of A. rank nullity (b) Find a basis for the nullspace of A. (c) Find a basis for the row space of A.
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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![(d) **Find a basis for the column space of A.**
The diagram shows a matrix with multiple columns, each consisting of spaces that represent entries in a matrix. The matrix is visually divided into three parts with green arrows indicating certain transformations or selections, suggesting these columns are part of the basis for the column space.
(e) **Determine whether or not the rows of A are linearly independent.**
- Independent
- Dependent
(f) **Let the columns of A be denoted by** \( \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4}, \) **and** \( \mathbf{a_5} \). **Which of the following sets is (are) linearly independent? (Select all that apply.)**
- [ ] \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_4} \} \)
- [ ] \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)
- [ ] \( \{ \mathbf{a_1}, \mathbf{a_3}, \mathbf{a_5} \} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8268c271-8991-49a2-aed9-a02bde5bd8ab%2Fecd481ca-1ade-49e6-a7e6-bf13b759e6bf%2Fqsrng9p_processed.png&w=3840&q=75)
Transcribed Image Text:(d) **Find a basis for the column space of A.**
The diagram shows a matrix with multiple columns, each consisting of spaces that represent entries in a matrix. The matrix is visually divided into three parts with green arrows indicating certain transformations or selections, suggesting these columns are part of the basis for the column space.
(e) **Determine whether or not the rows of A are linearly independent.**
- Independent
- Dependent
(f) **Let the columns of A be denoted by** \( \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}, \mathbf{a_4}, \) **and** \( \mathbf{a_5} \). **Which of the following sets is (are) linearly independent? (Select all that apply.)**
- [ ] \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_4} \} \)
- [ ] \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)
- [ ] \( \{ \mathbf{a_1}, \mathbf{a_3}, \mathbf{a_5} \} \)
![### Educational Exercise: Understanding Row-Equivalent Matrices
#### Problem Statement
Use the fact that matrices \( A \) and \( B \) are row-equivalent:
\[
A = \begin{bmatrix}
1 & 2 & 1 & 0 & 0 \\
2 & 5 & 1 & 1 & 0 \\
3 & 7 & 2 & 2 & -2 \\
5 & \textcolor{red}{11} & \textcolor{red}{4} & \textcolor{red}{-4} & \textcolor{red}{10}
\end{bmatrix}
\]
\[
B = \begin{bmatrix}
1 & 0 & 3 & 0 & -4 \\
0 & 1 & -1 & 0 & 2 \\
0 & 0 & 1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
#### Tasks
**(a) Find the rank and nullity of \( A \).**
- Rank: [Fill in the blank]
- Nullity: [Fill in the blank]
**(b) Find a basis for the nullspace of \( A \).**
A diagram is provided with two columns of empty boxes where solutions can be input.
**(c) Find a basis for the row space of \( A \).**
A diagram is provided with a 3x4 array of empty boxes where solutions can be input.
---
### Diagram Descriptions
The diagrams in parts (b) and (c) consist of sets of empty boxes. These boxes represent placeholders where the basis vectors for the respective spaces can be entered.
- In part (b), arrows indicate that there is a correspondence between the boxes, implying multiple bases vectors in the nullspace.
- In part (c), the array of boxes is organized to accommodate the row vectors that form the basis of the row space. Arrows suggest the direction in which entries should be considered.
This exercise helps in understanding the concepts of linear algebra related to row-equivalent matrices, their ranks, nulls, and spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8268c271-8991-49a2-aed9-a02bde5bd8ab%2Fecd481ca-1ade-49e6-a7e6-bf13b759e6bf%2Ffptn13_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Exercise: Understanding Row-Equivalent Matrices
#### Problem Statement
Use the fact that matrices \( A \) and \( B \) are row-equivalent:
\[
A = \begin{bmatrix}
1 & 2 & 1 & 0 & 0 \\
2 & 5 & 1 & 1 & 0 \\
3 & 7 & 2 & 2 & -2 \\
5 & \textcolor{red}{11} & \textcolor{red}{4} & \textcolor{red}{-4} & \textcolor{red}{10}
\end{bmatrix}
\]
\[
B = \begin{bmatrix}
1 & 0 & 3 & 0 & -4 \\
0 & 1 & -1 & 0 & 2 \\
0 & 0 & 1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
#### Tasks
**(a) Find the rank and nullity of \( A \).**
- Rank: [Fill in the blank]
- Nullity: [Fill in the blank]
**(b) Find a basis for the nullspace of \( A \).**
A diagram is provided with two columns of empty boxes where solutions can be input.
**(c) Find a basis for the row space of \( A \).**
A diagram is provided with a 3x4 array of empty boxes where solutions can be input.
---
### Diagram Descriptions
The diagrams in parts (b) and (c) consist of sets of empty boxes. These boxes represent placeholders where the basis vectors for the respective spaces can be entered.
- In part (b), arrows indicate that there is a correspondence between the boxes, implying multiple bases vectors in the nullspace.
- In part (c), the array of boxes is organized to accommodate the row vectors that form the basis of the row space. Arrows suggest the direction in which entries should be considered.
This exercise helps in understanding the concepts of linear algebra related to row-equivalent matrices, their ranks, nulls, and spaces.
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