Executing the following sage code constructs a matrix Q, then prints the matrix and its reduced row echelon form. Q=matrix(QQ, 4, 3, [-7, -22,-4, 12, 137, -13,-18, -213,21,15,115,-5]) print(Q) print() print(Q.rref()) The output is: -4] [ -7 -22 [ 12 137 -13] [-18 -213 21] [ 15 115 -5] [ [ [ 1 0 0 0 6/5] 1 -1/5] 0 0 0] 0] Are the columns of matrix Q linearly dependent, or linearly independent?
Executing the following sage code constructs a matrix Q, then prints the matrix and its reduced row echelon form. Q=matrix(QQ, 4, 3, [-7, -22,-4, 12, 137, -13,-18, -213,21,15,115,-5]) print(Q) print() print(Q.rref()) The output is: -4] [ -7 -22 [ 12 137 -13] [-18 -213 21] [ 15 115 -5] [ [ [ 1 0 0 0 6/5] 1 -1/5] 0 0 0] 0] Are the columns of matrix Q linearly dependent, or linearly independent?
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
![**Executing the following Sage code constructs a matrix Q, then prints the matrix and its reduced row echelon form.**
```sage
Q = matrix(QQ, 4, 3, [-7, -22, -4, 12, 137, -13, -18, -213, 21, 15, 115, -5])
print(Q)
print()
print(Q.rref())
```
**The output is:**
\[
\begin{bmatrix}
-7 & -22 & -4 \\
12 & 137 & -13 \\
-18 & -213 & 21 \\
15 & 115 & -5 \\
\end{bmatrix}
\]
\[
\begin{bmatrix}
1 & 0 & \frac{6}{5} \\
0 & 1 & -\frac{1}{5} \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
\]
**Are the columns of matrix Q linearly dependent, or linearly independent?**
The rows show the transformation to reduced row echelon form (RREF), where only two leading 1s indicate that the rank is less than the number of columns. This suggests the columns are linearly dependent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46eaea45-b11e-4a43-81b9-96bb1c4e5e25%2Ff5e3dac7-aca3-44eb-b798-61451536b8ab%2Frv9hlyg_processed.png&w=3840&q=75)
Transcribed Image Text:**Executing the following Sage code constructs a matrix Q, then prints the matrix and its reduced row echelon form.**
```sage
Q = matrix(QQ, 4, 3, [-7, -22, -4, 12, 137, -13, -18, -213, 21, 15, 115, -5])
print(Q)
print()
print(Q.rref())
```
**The output is:**
\[
\begin{bmatrix}
-7 & -22 & -4 \\
12 & 137 & -13 \\
-18 & -213 & 21 \\
15 & 115 & -5 \\
\end{bmatrix}
\]
\[
\begin{bmatrix}
1 & 0 & \frac{6}{5} \\
0 & 1 & -\frac{1}{5} \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
\]
**Are the columns of matrix Q linearly dependent, or linearly independent?**
The rows show the transformation to reduced row echelon form (RREF), where only two leading 1s indicate that the rank is less than the number of columns. This suggests the columns are linearly dependent.
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