15. Let A and b Show that the cquation Ax b does nol have a solution for all possible b, and describe the set of all b for which Ax=b does have a solution, 16. Repeat the requests from Exercise 15 wth
15. Let A and b Show that the cquation Ax b does nol have a solution for all possible b, and describe the set of all b for which Ax=b does have a solution, 16. Repeat the requests from Exercise 15 wth
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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![### Understanding Solutions of Linear Systems
#### 15. Analyzing Solutions for Specific Matrices and Vectors
Let \( A = \begin{bmatrix}
3 & -1 \\
9 & 3
\end{bmatrix} \) and \( b = \begin{bmatrix}
b_1 \\
b_2
\end{bmatrix} \). Show that the equation \( Ax = b \) does not have a solution for all possible \( b \), and describe the set of all \( b \) for which \( Ax = b \) **does** have a solution.
#### 16. Repetition with a Different Set of Matrices and Vectors
Repeat the requests from Exercise 15 with the following \( A \) and \( b \):
\[
A = \begin{bmatrix}
1 & 2 & -1 \\
2 & 2 & 0 \\
4 & 1 & 3
\end{bmatrix}, \quad
b = \begin{bmatrix}
b_1 \\
b_2 \\
b_3
\end{bmatrix}
\]
#### 17-20. Further Exploration with Matrices \( A \) and \( B \)
Exercises 17-20 refer to the matrices \( A \) and \( B \) below. Make appropriate calculations that justify your answers and mention an appropriate theorem.
\[
A = \begin{bmatrix}
1 & 3 & 0 & 3 \\
1 & 1 & 1 & 0 \\
0 & 4 & 2 & 8 \\
2 & 0 & 1 & 3
\end{bmatrix}, \quad
B = \begin{bmatrix}
1 & 4 & 2 \\
0 & 1 & 4 \\
0 & 2 & 6 \\
2 & 0 & 7
\end{bmatrix}
\]
##### 17. Pivot Positions in \( A \)
How many rows of \( A \) contain a pivot position? Does the equation \( Ax = b \) have a solution for each \( b \) in \( \mathbb{R}^4 \)?
##### 18. Linear Combinations and Spanning in \( \mathbb{R}^4 \)
Can every vector in \( \mathbb{R}^4 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2F5429d780-2aad-403b-843d-22c260b47f72%2F2hfdic.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Solutions of Linear Systems
#### 15. Analyzing Solutions for Specific Matrices and Vectors
Let \( A = \begin{bmatrix}
3 & -1 \\
9 & 3
\end{bmatrix} \) and \( b = \begin{bmatrix}
b_1 \\
b_2
\end{bmatrix} \). Show that the equation \( Ax = b \) does not have a solution for all possible \( b \), and describe the set of all \( b \) for which \( Ax = b \) **does** have a solution.
#### 16. Repetition with a Different Set of Matrices and Vectors
Repeat the requests from Exercise 15 with the following \( A \) and \( b \):
\[
A = \begin{bmatrix}
1 & 2 & -1 \\
2 & 2 & 0 \\
4 & 1 & 3
\end{bmatrix}, \quad
b = \begin{bmatrix}
b_1 \\
b_2 \\
b_3
\end{bmatrix}
\]
#### 17-20. Further Exploration with Matrices \( A \) and \( B \)
Exercises 17-20 refer to the matrices \( A \) and \( B \) below. Make appropriate calculations that justify your answers and mention an appropriate theorem.
\[
A = \begin{bmatrix}
1 & 3 & 0 & 3 \\
1 & 1 & 1 & 0 \\
0 & 4 & 2 & 8 \\
2 & 0 & 1 & 3
\end{bmatrix}, \quad
B = \begin{bmatrix}
1 & 4 & 2 \\
0 & 1 & 4 \\
0 & 2 & 6 \\
2 & 0 & 7
\end{bmatrix}
\]
##### 17. Pivot Positions in \( A \)
How many rows of \( A \) contain a pivot position? Does the equation \( Ax = b \) have a solution for each \( b \) in \( \mathbb{R}^4 \)?
##### 18. Linear Combinations and Spanning in \( \mathbb{R}^4 \)
Can every vector in \( \mathbb{R}^4 \)
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