What is the structure of the solution set for a consistent nonhomogeneous linear system Ax=b?
Conditions
- The system Ax=b is consistent.
- x₀ is a specific solution such that Ax₀=b.
Reasoning, step by step
- Find one particular solution x₀ to the equation Ax=b.
- Identify the kernel of A, which consists of all vectors z such that Az=0.
- Express the general solution as the sum of the particular solution and any vector from the kernel: ₀ + z.
- Recognize that this set forms an affine subspace shifted away from the origin if .
Example
The script states: 'If x₀ is one solution, all solutions have the form x₀+z with Az=0. A nonhomogeneous solution set is an affine translate of the kernel and need not pass through the origin.'
Common misconceptions
- Believing that the solution set always passes through the origin.
- Thinking that the kernel itself is the solution set for nonhomogeneous systems.
- Confusing the particular solution with the general solution.
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