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Math Help - Prove if a linear system has more than 1 solution, it has infinite solutions

  1. #1
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    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.
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  2. #2
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    Consider:  A \left[X_1 + c\left(X_2 - X_1\right)\right]

    = AX_1 + cA \left(X_2 - X_1\right)

    = AX_1 + cAX_2 - cAX_1

    = \cdots

    = B

    ___________

    To show that all solutions are different for all c, suppose that there exists two different c_1 and c_2 such that:
    X_1 + c_1 \left(X_2 - X_1\right) = X_1 + c_2 \left(X_2 - X_1\right)

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