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Math Help - Gauss' Method Proof

  1. #1
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    Gauss' Method Proof

    This is a really basic question. I want to prove the following statment:

    If a linear system is changed to another by multiplying both sides of an equation by a nonzero constant, then the two systems have the same solution.

    I know this is very obvious and seemingly pointless to prove. However, I never have been able to tackle the concept of proof and I can rarely figure out how to begin on things like this.

    This is my linear system:

    a_{1,1}x_1+a_{1,2}x_2+...+a_{1,n}x_n=d_1

    a_{2,1}x_1+a_{2,2}x_2+...+a_{2,n}x_n=d_2


    a_{m,1}x_1+a_{m,2}x_2+...+a_{m,n}x_n=d_m

    The solution set is an n-tuple {s_1,s_2....s_n}. Where would I go from here?
    Last edited by adkinsjr; November 9th 2009 at 02:54 AM.
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  2. #2
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    I wrote out an attempt to prove thie statement in my original post.

    a_{1,1}x_1+a_{1,2}x_2+...+a_{1,n}x_n=d_1

    a_{2,1}x_1+a_{2,2}x_2+...+a_{2,n}x_n=d_2


    a_{m,1}x_1+a_{m,2}x_2+...+a_{m,n}x_n=d_m

    The solution set is an n-tuple {s_1,s_2....s_n}.

    Since the n-tuple is a solution to the linear system, I can choose the linear combination a_{2,1}x_1+a_{2,2}x_2+...+a_{2,n}x_n=d_2 and write the true statement a_{2,1}S_1+a_{2,2}S_2+...+a_{2,n}S_n=d_2. Multiplying both sides by a nonzero constant C gives Ca_{2,1}S_1+Ca_{2,2}S_2+...+Ca_{2,n}S_n=Cd_2. But this is just the requirement for the linear combination Ca_{2,1}x_1+Ca_{2,2}x_2+...+Ca_{2,n}x_n=Cd_2 to have a solution set S_1,...,S_n. Therefore, the solution set is unchanged.
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