2. Consider the following matrix and its RREF [40-8-1 6 3 3 A = 3 1 -2 4 1 1 0 0 3 -1 0 2 43 4 -2-9 30-6 06 Explain all your answers (a) Determine a basis for Col(A) (b) Determine a basis for Nul(A). (c) Determine rank(A) and nullity(A). RREF 1 0-2 0 0 1 0 00 01 00 00 00 74000 28410 01 -3 0 0 1 0
2. Consider the following matrix and its RREF [40-8-1 6 3 3 A = 3 1 -2 4 1 1 0 0 3 -1 0 2 43 4 -2-9 30-6 06 Explain all your answers (a) Determine a basis for Col(A) (b) Determine a basis for Nul(A). (c) Determine rank(A) and nullity(A). RREF 1 0-2 0 0 1 0 00 01 00 00 00 74000 28410 01 -3 0 0 1 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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Question
![### Matrix and RREF Analysis
Consider the following matrix \( A \) and its Reduced Row Echelon Form (RREF):
\[
A = \left[\begin{array}{cccccc}
4 & 0 & -8 & -1 & 4 & -4 \\
3 & 3 & 6 & 1 & 1 & 0 \\
3 & 1 & -2 & 0 & 3 & -1 \\
4 & 3 & 4 & -2 & -9 & 0 \\
3 & 0 & -6 & 0 & 6 & 2 \\
\end{array}\right]
\xrightarrow{\text{RREF}}
\left[\begin{array}{cccccc}
1 & 0 & -2 & 0 & 2 & 0 \\
0 & 1 & 4 & 0 & -3 & 0 \\
0 & 0 & 0 & 1 & 4 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}\right]
\]
#### Explain all your answers
**(a) Determine a basis for \(\text{Col}(A)\).**
To find a basis for the column space \(\text{Col}(A)\), note the columns in the original matrix \( A \) that correspond to the leading 1s in the RREF. These are the first, second, fourth, and sixth columns. Thus, a basis for \(\text{Col}(A)\) comprises these columns from \( A \).
**(b) Determine a basis for \(\text{Nul}(A)\).**
To find a basis for the null space \(\text{Nul}(A)\), solve the homogenous system corresponding to the RREF of \( A \). Set the free variables and express other variables in terms of these. The solution vector that results gives a basis for \(\text{Nul}(A)\).
**(c) Determine \(\text{rank}(A)\) and \(\text{nullity}(A)\).**
- \(\text{rank}(A)\) is the number of leading 1s in the RREF, which is 4.
- \(\text{nullity](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F992f5037-f1cc-4483-8aa3-731b184958ad%2F701ba778-84ad-4884-9dd8-bd85d562d63e%2F2kdlcqa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix and RREF Analysis
Consider the following matrix \( A \) and its Reduced Row Echelon Form (RREF):
\[
A = \left[\begin{array}{cccccc}
4 & 0 & -8 & -1 & 4 & -4 \\
3 & 3 & 6 & 1 & 1 & 0 \\
3 & 1 & -2 & 0 & 3 & -1 \\
4 & 3 & 4 & -2 & -9 & 0 \\
3 & 0 & -6 & 0 & 6 & 2 \\
\end{array}\right]
\xrightarrow{\text{RREF}}
\left[\begin{array}{cccccc}
1 & 0 & -2 & 0 & 2 & 0 \\
0 & 1 & 4 & 0 & -3 & 0 \\
0 & 0 & 0 & 1 & 4 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}\right]
\]
#### Explain all your answers
**(a) Determine a basis for \(\text{Col}(A)\).**
To find a basis for the column space \(\text{Col}(A)\), note the columns in the original matrix \( A \) that correspond to the leading 1s in the RREF. These are the first, second, fourth, and sixth columns. Thus, a basis for \(\text{Col}(A)\) comprises these columns from \( A \).
**(b) Determine a basis for \(\text{Nul}(A)\).**
To find a basis for the null space \(\text{Nul}(A)\), solve the homogenous system corresponding to the RREF of \( A \). Set the free variables and express other variables in terms of these. The solution vector that results gives a basis for \(\text{Nul}(A)\).
**(c) Determine \(\text{rank}(A)\) and \(\text{nullity}(A)\).**
- \(\text{rank}(A)\) is the number of leading 1s in the RREF, which is 4.
- \(\text{nullity
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