Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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For the matrix given find
1. It’s column space.
2.It’s null space.
3.It’s row space.
![For this educational website content, we will be discussing matrix transformations and provide a detailed transcription and explanation of the given matrix problem.
---
### Matrix Transformation Example
We are given a matrix \( A \), and our goal is to find the reduced row echelon form of \( A \).
The matrix \( A \) is as follows:
\[
A = \begin{pmatrix}
-1 & 3 & 1 \\
2 & 2 & 4 \\
-6 & -2 & -1 \\
-8 & -1 & 9 \\
1 & -1 & -1
\end{pmatrix}
\]
To solve for the reduced row echelon form (RREF) of matrix \( A \), follow the Gaussian elimination steps which involve row operations such as:
1. Swapping rows
2. Multiplying a row by a non-zero scalar
3. Adding or subtracting multiples of one row to another row
---
#### Detailed Steps (Note: Example steps provided without exact intermediate results)
1. **Row Operations:**
- Identify the leading entry in each row.
- Ensure each leading entry is 1 (if necessary, scale the row).
- Use row operations to create zeros in all positions below and above each leading entry.
2. **Example Steps:**
- Scale Row 1, if needed, so the first element is 1.
- Create zeros below the first element of Row 1.
- Move to the second row and scale so the second element is 1.
- Eliminate elements below and above this leading 1.
- Continue for subsequent rows.
\[
\begin{pmatrix}
1 & a & b \\
0 & 1 & c \\
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]
(Note: \(a\), \(b\), and \(c\) represent placeholder numbers after performing row operations corresponding to their respective positions).
3. **Final Matrix:**
The final matrix should be in the form where the leading coefficient of each row is 1, and all other elements in the column containing the leading coefficient are zeros. This is achieved after all necessary row operations are performed.
---
### Conclusion
Performing these steps results in the reduced row echelon form of the](https://content.bartleby.com/qna-images/question/be52e616-7bd7-4d85-a476-7e5fcec53ee5/70fe694c-3646-42e2-bf65-7e2fd7a7e942/0b8md8g_processed.jpeg)
Transcribed Image Text:For this educational website content, we will be discussing matrix transformations and provide a detailed transcription and explanation of the given matrix problem.
---
### Matrix Transformation Example
We are given a matrix \( A \), and our goal is to find the reduced row echelon form of \( A \).
The matrix \( A \) is as follows:
\[
A = \begin{pmatrix}
-1 & 3 & 1 \\
2 & 2 & 4 \\
-6 & -2 & -1 \\
-8 & -1 & 9 \\
1 & -1 & -1
\end{pmatrix}
\]
To solve for the reduced row echelon form (RREF) of matrix \( A \), follow the Gaussian elimination steps which involve row operations such as:
1. Swapping rows
2. Multiplying a row by a non-zero scalar
3. Adding or subtracting multiples of one row to another row
---
#### Detailed Steps (Note: Example steps provided without exact intermediate results)
1. **Row Operations:**
- Identify the leading entry in each row.
- Ensure each leading entry is 1 (if necessary, scale the row).
- Use row operations to create zeros in all positions below and above each leading entry.
2. **Example Steps:**
- Scale Row 1, if needed, so the first element is 1.
- Create zeros below the first element of Row 1.
- Move to the second row and scale so the second element is 1.
- Eliminate elements below and above this leading 1.
- Continue for subsequent rows.
\[
\begin{pmatrix}
1 & a & b \\
0 & 1 & c \\
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]
(Note: \(a\), \(b\), and \(c\) represent placeholder numbers after performing row operations corresponding to their respective positions).
3. **Final Matrix:**
The final matrix should be in the form where the leading coefficient of each row is 1, and all other elements in the column containing the leading coefficient are zeros. This is achieved after all necessary row operations are performed.
---
### Conclusion
Performing these steps results in the reduced row echelon form of the
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