Final answer:
To show that if x₂ and x₁ are solutions to the linear system Ax=b, then x₂-x₁ is a solution to the associated homogeneous system Ax=0, we need to prove that the difference of the solutions, x₂-x₁, satisfies the equation Ax=0.
Step-by-step explanation:
To show that if x₂ and x₂ are solutions to the linear system Ax=b, then x₂-x₁ is a solution to the associated homogeneous system Ax=0, we need to prove that the difference of the solutions, x₂-x₁, satisfies the equation Ax=0.
Let's assume that x₂ and x₁ are solutions to Ax=b. This means that when we substitute x₂ and x₁ into the equation Ax=b, we get the same value on both sides of the equation.
Now, subtract the equation when x₁ is substituted from the equation when x₂ is substituted: Ax₂ - Ax₁ = b - b. Since the two sides of the equation are equal, this simplifies to Ax₂ - Ax₁ = 0. Therefore, x₂-x₁ is a solution to the associated homogeneous system Ax=0.