en matrix A, find k such that Nul A is a subspace of rk and find m such that Col A is a subspace of m . 1 -2 0 6 -4 5 -1 -3 3-5 A = A) k= 5, m = 5 B) K = 5, m = 2 C) k = 2, m = 2 D) k = 2, m = 5 3)
en matrix A, find k such that Nul A is a subspace of rk and find m such that Col A is a subspace of m . 1 -2 0 6 -4 5 -1 -3 3-5 A = A) k= 5, m = 5 B) K = 5, m = 2 C) k = 2, m = 2 D) k = 2, m = 5 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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Question
![### Problem 3
For the given matrix \( A \), find \( k \) such that \( \text{Nul } A \) is a subspace of \( \mathbb{R}^k \) and find \( m \) such that \( \text{Col } A \) is a subspace of \( \mathbb{R}^m \).
\[
A = \begin{bmatrix}
1 & -2 \\
0 & 6 \\
-4 & 5 \\
-1 & -3 \\
3 & -5
\end{bmatrix}
\]
Options:
- A) \( k = 5, m = 5 \)
- B) \( k = 5, m = 2 \)
- C) \( k = 2, m = 2 \)
- D) \( k = 2, m = 5 \)
### Problem 4
For the given matrix \( A \), find \( k \) such that \( \text{Nul } A \) is a subspace of \( \mathbb{R}^k \) and find \( m \) such that \( \text{Col } A \) is a subspace of \( \mathbb{R}^m \).
\[
A = \begin{bmatrix}
4 & 0 & 0 & -1 & 1 & -7 \\
2 & 6 & -5 & -1 & 0 & 3 \\
-3 & -4 & 4 & -5 & 5 & -3
\end{bmatrix}
\]
Options:
- A) \( k = 3, m = 3 \)
- B) \( k = 6, m = 3 \)
- C) \( k = 3, m = 6 \)
- D) \( k = 6, m = 6 \)
### Problem 5
Determine if the vector \( \mathbf{u} \) is in the column space of matrix \( A \) and whether it is in the null space of \( A \).
\[
\mathbf{u} = \begin{bmatrix}
-2 \\
-5 \\
-2
\end{bmatrix}, \quad A = \begin{bmatrix}
1 & -3 & 4 \\
-1 & 0 &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8aa5a8f7-80ac-4340-a2ba-c822c7404ea9%2F4ab03174-999f-40aa-b4f5-cc91561c2cd9%2F063211h_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3
For the given matrix \( A \), find \( k \) such that \( \text{Nul } A \) is a subspace of \( \mathbb{R}^k \) and find \( m \) such that \( \text{Col } A \) is a subspace of \( \mathbb{R}^m \).
\[
A = \begin{bmatrix}
1 & -2 \\
0 & 6 \\
-4 & 5 \\
-1 & -3 \\
3 & -5
\end{bmatrix}
\]
Options:
- A) \( k = 5, m = 5 \)
- B) \( k = 5, m = 2 \)
- C) \( k = 2, m = 2 \)
- D) \( k = 2, m = 5 \)
### Problem 4
For the given matrix \( A \), find \( k \) such that \( \text{Nul } A \) is a subspace of \( \mathbb{R}^k \) and find \( m \) such that \( \text{Col } A \) is a subspace of \( \mathbb{R}^m \).
\[
A = \begin{bmatrix}
4 & 0 & 0 & -1 & 1 & -7 \\
2 & 6 & -5 & -1 & 0 & 3 \\
-3 & -4 & 4 & -5 & 5 & -3
\end{bmatrix}
\]
Options:
- A) \( k = 3, m = 3 \)
- B) \( k = 6, m = 3 \)
- C) \( k = 3, m = 6 \)
- D) \( k = 6, m = 6 \)
### Problem 5
Determine if the vector \( \mathbf{u} \) is in the column space of matrix \( A \) and whether it is in the null space of \( A \).
\[
\mathbf{u} = \begin{bmatrix}
-2 \\
-5 \\
-2
\end{bmatrix}, \quad A = \begin{bmatrix}
1 & -3 & 4 \\
-1 & 0 &
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