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Is the following statement true?

If an inhomogeneous system of linear equations has several possible solutions, then the associated homogeneous system of linear equations also has several solutions.

My initial intuition would be that this is correct, but I can't show it formally. How would one prove or disprove this statement?

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  • $\begingroup$ Just subtract the two systems of equations. If $Ax_1=b$ and $Ax_2=b$, with $x_1\neq x_2$, then $A(x_1-x_2)=Ax_1-Ax_2=b-b=0$. Therefore, $x_1-x_2$, which is not the zero vector, is a solution of $Ax=0$. So, at least $0$ and $x_1-x_2$ are two solutions of the homogeneous system. $\endgroup$
    – plop
    Sep 6, 2022 at 15:25
  • $\begingroup$ @yousafe007 If you are asking about a linear system of equations, then please make this explicit in your question. $\endgroup$ Sep 6, 2022 at 16:10

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