Consider the matrices 2 1 4 0 13 1 2 -1 0 3 -6 1 |-1 2 -7 1 4 1 17 -1 7 [1 0 3 7 0 8 25 -3 -19 1 -2 -1 0 0 R = 0 0 0 0 0 0 0 0 1 9 0 0 0 0 A = 1 1 9. and 1 4. -5 2 10 2 1 2 34 You may assume that R is the reduced row-echelon form of A. Let ã1, ...,ā7 E R° denote the columns of A, in left-to-right order. (a) Which of the vectors ā3, ā4, and ās belong to the span of ã1 and ã2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ã1,...,ā7? No justification necessary. (c) Does āz belong to the span of ã1, ã2, and ã4? If not, explain why not; if so, write ãg explicitly as a linear combination of ā1, ā2, and ã4.
Consider the matrices 2 1 4 0 13 1 2 -1 0 3 -6 1 |-1 2 -7 1 4 1 17 -1 7 [1 0 3 7 0 8 25 -3 -19 1 -2 -1 0 0 R = 0 0 0 0 0 0 0 0 1 9 0 0 0 0 A = 1 1 9. and 1 4. -5 2 10 2 1 2 34 You may assume that R is the reduced row-echelon form of A. Let ã1, ...,ā7 E R° denote the columns of A, in left-to-right order. (a) Which of the vectors ā3, ā4, and ās belong to the span of ã1 and ã2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ã1,...,ā7? No justification necessary. (c) Does āz belong to the span of ã1, ã2, and ã4? If not, explain why not; if so, write ãg explicitly as a linear combination of ā1, ā2, and ã4.
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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**4. Consider the matrices**
\[ A = \begin{bmatrix}
2 & 4 & 0 & 13 & -1 & 7 \\
1 & 2 & 5 & 25 & -3 & -19 \\
0 & 3 & -6 & 1 & 1 & 9 \\
2 & -7 & -1 & -5 & 2 & 10 \\
1 & 4 & 1 & 17 & 2 & 34
\end{bmatrix} \]
and
\[ R = \begin{bmatrix}
1 & 0 & 3 & 0 & 7 & 0 & 8 \\
0 & 1 & -2 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 4 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 9 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \]
You may assume that \( R \) is the reduced row-echelon form of \( A \). Let \(\mathbf{a}_1, \ldots, \mathbf{a}_7 \in \mathbb{R}^5\) denote the columns of \( A \), in left-to-right order.
(a) Which of the vectors \(\mathbf{a}_3, \mathbf{a}_4,\) and \(\mathbf{a}_5\) belong to the span of \(\mathbf{a}_1\) and \(\mathbf{a}_2\)? No justification necessary.
(b) What is the largest number of linearly independent vectors that can be chosen from among \(\mathbf{a}_1, \ldots, \mathbf{a}_7\)? No justification necessary.
(c) Does \(\mathbf{a}_5\) belong to the span of \(\mathbf{a}_1, \mathbf{a}_2,\) and \(\mathbf{a}_4\)? If not, explain why not; if so, write \(\mathbf{a}_5\) explicitly as a linear combination of \(\mathbf{a}_1, \mathbf{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69a98968-5f96-40b4-b8f7-c8cceea98cdb%2Fda488b29-06f9-4b03-a5c2-fc272aea9039%2Fhwylwy7_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcribed text suitable for an educational website:
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**4. Consider the matrices**
\[ A = \begin{bmatrix}
2 & 4 & 0 & 13 & -1 & 7 \\
1 & 2 & 5 & 25 & -3 & -19 \\
0 & 3 & -6 & 1 & 1 & 9 \\
2 & -7 & -1 & -5 & 2 & 10 \\
1 & 4 & 1 & 17 & 2 & 34
\end{bmatrix} \]
and
\[ R = \begin{bmatrix}
1 & 0 & 3 & 0 & 7 & 0 & 8 \\
0 & 1 & -2 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 4 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 9 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \]
You may assume that \( R \) is the reduced row-echelon form of \( A \). Let \(\mathbf{a}_1, \ldots, \mathbf{a}_7 \in \mathbb{R}^5\) denote the columns of \( A \), in left-to-right order.
(a) Which of the vectors \(\mathbf{a}_3, \mathbf{a}_4,\) and \(\mathbf{a}_5\) belong to the span of \(\mathbf{a}_1\) and \(\mathbf{a}_2\)? No justification necessary.
(b) What is the largest number of linearly independent vectors that can be chosen from among \(\mathbf{a}_1, \ldots, \mathbf{a}_7\)? No justification necessary.
(c) Does \(\mathbf{a}_5\) belong to the span of \(\mathbf{a}_1, \mathbf{a}_2,\) and \(\mathbf{a}_4\)? If not, explain why not; if so, write \(\mathbf{a}_5\) explicitly as a linear combination of \(\mathbf{a}_1, \mathbf{
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