Thread: Prove if a linear system has more than 1 solution, it has infinite solutions

"Hint: Show that if X1 and X2 are different solutions to AX = B, then X1 + c(X2 - X1) is also a solution, for every real number c. Also, show that these solutions are different."

I think I know how to show they are different. If X2 is different from X1 then X2 - X1 =/= 0. Therefore X1 + c(X2 -X1) =/= X1. (except for the one case when c = 0)

However I don't know how to prove that X1 + c(X2 - X1) is also a solution.

Consider:









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To show that all solutions are different for all , suppose that there exists two different and such that:


Now show that they must be equal, leading to a contradiction/