Exercises 9-12 display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. 9. A 1 -3 -3 2-4 9 -1 5 4 -3 7 2-6 -4 12 2 N 1-3 2-4 05-7 0 0 0 00 5 00 0
Exercises 9-12 display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. 9. A 1 -3 -3 2-4 9 -1 5 4 -3 7 2-6 -4 12 2 N 1-3 2-4 05-7 0 0 0 00 5 00 0
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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![**Exercises 9–12** display a matrix \( A \) and an echelon form of \( A \). Find bases for \( \text{Col} A \) and \( \text{Nul} A \), and then state the dimensions of these subspaces.
**9.**
\[
A = \begin{bmatrix}
1 & -3 & 2 & -4 \\
-3 & 9 & -1 & 5 \\
2 & -6 & 4 & -3 \\
-4 & 12 & 2 & 7
\end{bmatrix}
\quad \sim \quad
\begin{bmatrix}
1 & -3 & 2 & -4 \\
0 & 0 & 5 & -7 \\
0 & 0 & 0 & 5 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]
**10.**
\[
A = \begin{bmatrix}
1 & -2 & 9 & 5 & 4 \\
-1 & -1 & 6 & 5 & -3 \\
2 & 0 & -6 & -1 & 2 \\
4 & 1 & 9 & 1 & -9
\end{bmatrix}
\quad \sim \quad
\begin{bmatrix}
1 & -2 & 9 & 5 & 4 \\
0 & 1 & -3 & 0 & -7 \\
0 & 0 & 0 & 1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
**Details:**
- For each matrix \( A \) provided, the echelon form is computed and displayed alongside the original matrix.
- Follow the steps to convert each original matrix \( A \) to its echelon form.
- Identify the pivot columns in the echelon form; these columns form the basis for \( \text{Col} A \).
- Solve the homogeneous system \( A\mathbf{x} = 0 \) to find the basis for \( \text{Nul} A \).
**Column Space (Col A):**
The basis for the column space consists of the pivot columns of the original matrix \( A \).
**Null Space (Nul A):**
The basis for the null](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2Fff5cb74a-f116-47bc-98b9-07a290e24a7b%2F1seprol_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercises 9–12** display a matrix \( A \) and an echelon form of \( A \). Find bases for \( \text{Col} A \) and \( \text{Nul} A \), and then state the dimensions of these subspaces.
**9.**
\[
A = \begin{bmatrix}
1 & -3 & 2 & -4 \\
-3 & 9 & -1 & 5 \\
2 & -6 & 4 & -3 \\
-4 & 12 & 2 & 7
\end{bmatrix}
\quad \sim \quad
\begin{bmatrix}
1 & -3 & 2 & -4 \\
0 & 0 & 5 & -7 \\
0 & 0 & 0 & 5 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]
**10.**
\[
A = \begin{bmatrix}
1 & -2 & 9 & 5 & 4 \\
-1 & -1 & 6 & 5 & -3 \\
2 & 0 & -6 & -1 & 2 \\
4 & 1 & 9 & 1 & -9
\end{bmatrix}
\quad \sim \quad
\begin{bmatrix}
1 & -2 & 9 & 5 & 4 \\
0 & 1 & -3 & 0 & -7 \\
0 & 0 & 0 & 1 & -2 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
**Details:**
- For each matrix \( A \) provided, the echelon form is computed and displayed alongside the original matrix.
- Follow the steps to convert each original matrix \( A \) to its echelon form.
- Identify the pivot columns in the echelon form; these columns form the basis for \( \text{Col} A \).
- Solve the homogeneous system \( A\mathbf{x} = 0 \) to find the basis for \( \text{Nul} A \).
**Column Space (Col A):**
The basis for the column space consists of the pivot columns of the original matrix \( A \).
**Null Space (Nul A):**
The basis for the null
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