Civen that A and B are row equivalent. -1/6 100 -1/6 0 1 0 4 -6 3 1 -3/2 0 0 0 2 A = -2 -3 -2 0 3 1 0. B = 1/3 0 0 1 3 1 1 -8 12 -8 0 -2 2 1/3 (a) Find the rank of A. (b) Find a basis for the row space of A. (c) Find a basis for the column space of A.
Civen that A and B are row equivalent. -1/6 100 -1/6 0 1 0 4 -6 3 1 -3/2 0 0 0 2 A = -2 -3 -2 0 3 1 0. B = 1/3 0 0 1 3 1 1 -8 12 -8 0 -2 2 1/3 (a) Find the rank of A. (b) Find a basis for the row space of A. (c) Find a basis for the column 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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![Given that \( A \) and \( B \) are row equivalent:
\[ A = \begin{bmatrix} 4 & -6 & 0 & 3 & 5 & 2 \\ 2 & -3 & -2 & 0 & 3 & 1 \\ -2 & 3 & 2 & 1 & 2 & 1 \\ -8 & 12 & -8 & 0 & -2 & 2 \end{bmatrix} \]
\[ B = \begin{bmatrix} 1 & -\frac{3}{2} & 0 & 0 & 0 & -\frac{1}{6} \\ 0 & 0 & 1 & 0 & 0 & -\frac{1}{6} \\ 0 & 0 & 0 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & 0 & 0 & 1 & \frac{1}{3} \end{bmatrix} \]
**Questions:**
(a) Find the rank of \( A \).
(b) Find a basis for the row space of \( A \).
(c) Find a basis for the column space of \( A \).
(d) Find a basis for the nullspace of \( A \).
(e) Are the first 4 columns of \( A \) linearly independent? Explain your answer.
(f) Is the last column of \( A \) in the span of column 1, 3, 4, and 5? Explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff57fbb07-e1bd-418f-9d01-1252f5b1cb71%2F962d5c13-f5e9-42c8-a089-c11747a37368%2Fl963u8h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given that \( A \) and \( B \) are row equivalent:
\[ A = \begin{bmatrix} 4 & -6 & 0 & 3 & 5 & 2 \\ 2 & -3 & -2 & 0 & 3 & 1 \\ -2 & 3 & 2 & 1 & 2 & 1 \\ -8 & 12 & -8 & 0 & -2 & 2 \end{bmatrix} \]
\[ B = \begin{bmatrix} 1 & -\frac{3}{2} & 0 & 0 & 0 & -\frac{1}{6} \\ 0 & 0 & 1 & 0 & 0 & -\frac{1}{6} \\ 0 & 0 & 0 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & 0 & 0 & 1 & \frac{1}{3} \end{bmatrix} \]
**Questions:**
(a) Find the rank of \( A \).
(b) Find a basis for the row space of \( A \).
(c) Find a basis for the column space of \( A \).
(d) Find a basis for the nullspace of \( A \).
(e) Are the first 4 columns of \( A \) linearly independent? Explain your answer.
(f) Is the last column of \( A \) in the span of column 1, 3, 4, and 5? Explain your answer.
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