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Suppose Ax = b, where b is not the zero vector, is a consistent linear system. Suppose that u is a solution to the nonhomogeneous system and v is a solution to the associated homogeneous system Ax = 0. Prove that u + v is a solution to Ax = b and that every solution x to the nonhomogeneous system Ax = b can be written as x₂ + xh, where xp is a particular solution to the nonhomogeneous system and x is a solution to the associated homogeneous system.

Suppose Ax = b, where b is not the zero vector, is a consistent linear system. Suppose that u is a solution to the nonhomogeneous system and v is a solution to the associated homogeneous system Ax = 0. Prove that u + v is a solution to Ax = b and that every solution x to the nonhomogeneous system Ax = b can be written as x₂ + xh, where xp is a particular solution to the nonhomogeneous system and x is a solution to the associated homogeneous system.

Advanced Engineering Mathematics
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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### Transcription for Educational Website: --- #### Linear Algebra: Solutions to Homogeneous and Nonhomogeneous Systems Suppose \( A\mathbf{x} = \mathbf{b} \), where \( \mathbf{b} \) is not the zero vector, is a consistent linear system. Suppose that \( \mathbf{u} \) is a solution to the nonhomogeneous system and \( \mathbf{v} \) is a solution to the associated homogeneous system \( A\mathbf{x} = \mathbf{0} \). Prove that \( \mathbf{u} + \mathbf{v} \) is a solution to \( A\mathbf{x} = \mathbf{b} \) and that every solution \( \mathbf{x} \) to the nonhomogeneous system \( A\mathbf{x} = \mathbf{b} \) can be written as \( \mathbf{x}_p + \mathbf{x}_h \), where \( \mathbf{x}_p \) is a particular solution to the nonhomogeneous system and \( \mathbf{x}_h \) is a solution to the associated homogeneous system. --- In this problem, we are working with both nonhomogeneous and homogeneous linear systems. The key goals are to demonstrate the properties of these systems and solutions: 1. **Combined Solution**: - If \( \mathbf{u} \) is a solution to the nonhomogeneous system \( A\mathbf{u} = \mathbf{b} \), and \( \mathbf{v} \) is a solution to the homogeneous system \( A\mathbf{v} = \mathbf{0} \), we want to prove that their sum \( \mathbf{u} + \mathbf{v} \) is also a solution to the nonhomogeneous system \( A\mathbf{x} = \mathbf{b} \). 2. **General Solution Form**: - Any solution \( \mathbf{x} \) to the nonhomogeneous system can be expressed as the sum of a particular solution \( \mathbf{x}_p \) and a solution \( \mathbf{x}_h \) to the homogeneous system. To effectively navigate through the problem, we will delve into the following proofs: - **Proof of Combined Solution**: - Starting with \( A\mathbf{u} = \mathbf{b} \) and \( A\mathbf{v} = \mathbf{0} \), show that \(
Transcribed Image Text:### Transcription for Educational Website: --- #### Linear Algebra: Solutions to Homogeneous and Nonhomogeneous Systems Suppose \( A\mathbf{x} = \mathbf{b} \), where \( \mathbf{b} \) is not the zero vector, is a consistent linear system. Suppose that \( \mathbf{u} \) is a solution to the nonhomogeneous system and \( \mathbf{v} \) is a solution to the associated homogeneous system \( A\mathbf{x} = \mathbf{0} \). Prove that \( \mathbf{u} + \mathbf{v} \) is a solution to \( A\mathbf{x} = \mathbf{b} \) and that every solution \( \mathbf{x} \) to the nonhomogeneous system \( A\mathbf{x} = \mathbf{b} \) can be written as \( \mathbf{x}_p + \mathbf{x}_h \), where \( \mathbf{x}_p \) is a particular solution to the nonhomogeneous system and \( \mathbf{x}_h \) is a solution to the associated homogeneous system. --- In this problem, we are working with both nonhomogeneous and homogeneous linear systems. The key goals are to demonstrate the properties of these systems and solutions: 1. **Combined Solution**: - If \( \mathbf{u} \) is a solution to the nonhomogeneous system \( A\mathbf{u} = \mathbf{b} \), and \( \mathbf{v} \) is a solution to the homogeneous system \( A\mathbf{v} = \mathbf{0} \), we want to prove that their sum \( \mathbf{u} + \mathbf{v} \) is also a solution to the nonhomogeneous system \( A\mathbf{x} = \mathbf{b} \). 2. **General Solution Form**: - Any solution \( \mathbf{x} \) to the nonhomogeneous system can be expressed as the sum of a particular solution \( \mathbf{x}_p \) and a solution \( \mathbf{x}_h \) to the homogeneous system. To effectively navigate through the problem, we will delve into the following proofs: - **Proof of Combined Solution**: - Starting with \( A\mathbf{u} = \mathbf{b} \) and \( A\mathbf{v} = \mathbf{0} \), show that \(
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