4.2.8. Consider a linear system of equations Ax = b for which y¹b = 0 for every y EN (AT). Explain why this means the system must be consistent. Solution:
4.2.8. Consider a linear system of equations Ax = b for which y¹b = 0 for every y EN (AT). Explain why this means the system must be consistent. Solution:
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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![### Topic: Consistency of a Linear System
#### Problem 4.2.8
Consider a linear system of equations \( Ax = b \) for which \( \mathbf{y}^T \mathbf{b} = 0 \) for every \( \mathbf{y} \in N(A^T) \). Explain why this means the system must be consistent.
#### Solution
Given:
\[ \mathbf{y}^T \mathbf{b} = 0 \quad \forall \, \mathbf{y} \in N(A^T) = R(P_2^T) \]
This implies:
\[ P_2 \mathbf{b} = 0 \]
Therefore:
\[ \mathbf{b} \in N(P_2) = R(A) \]
### Explanation
The solution states that for the linear system \( Ax = b \) to be consistent, vector \( \mathbf{b} \) must be in the range of matrix \( A \) (denoted \( R(A) \)). The condition \( \mathbf{y}^T \mathbf{b} = 0 \) for all \( \mathbf{y} \in N(A^T) \) ensures that \( \mathbf{b} \) is orthogonal to the null space of \( A^T \), implying \( \mathbf{b} \) lies in the row space of \( A \) (denoted \( R(A) \)). This makes the system consistent since \( \mathbf{b} \) can be expressed as a linear combination of the columns of \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98b6e310-08ba-4e1d-a9bc-704b45d2ce6c%2F482fb45e-27cb-4a15-af85-bdf5a5e5dec0%2Fv6ufx8_processed.png&w=3840&q=75)
Transcribed Image Text:### Topic: Consistency of a Linear System
#### Problem 4.2.8
Consider a linear system of equations \( Ax = b \) for which \( \mathbf{y}^T \mathbf{b} = 0 \) for every \( \mathbf{y} \in N(A^T) \). Explain why this means the system must be consistent.
#### Solution
Given:
\[ \mathbf{y}^T \mathbf{b} = 0 \quad \forall \, \mathbf{y} \in N(A^T) = R(P_2^T) \]
This implies:
\[ P_2 \mathbf{b} = 0 \]
Therefore:
\[ \mathbf{b} \in N(P_2) = R(A) \]
### Explanation
The solution states that for the linear system \( Ax = b \) to be consistent, vector \( \mathbf{b} \) must be in the range of matrix \( A \) (denoted \( R(A) \)). The condition \( \mathbf{y}^T \mathbf{b} = 0 \) for all \( \mathbf{y} \in N(A^T) \) ensures that \( \mathbf{b} \) is orthogonal to the null space of \( A^T \), implying \( \mathbf{b} \) lies in the row space of \( A \) (denoted \( R(A) \)). This makes the system consistent since \( \mathbf{b} \) can be expressed as a linear combination of the columns of \( A \).
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