Prove that if more than one solution exists, then....

[buckaroobill]

hi, i'm practicing proofs for an upcoming test regarding systems of linear equations, matrices, and Gaussian elimination. i was wondering if anyone could show me how to do the following that i found in my textbook in case i'm asked to do a similar one:

Prove that if more than one solution in a system of linear equations exists, then an infinite number of solutions exists. (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).

 

[mark07]

Let's just follow the hint. Let x1 and x2 be two different solutions. Then look at what happens when you multiply x1 + c(x2 - x1) by A (use linearity of matrix multiplication). Are you getting B? What does that mean? After that, let c1 and c2 be two different non-zero real numbers. Then, it is easy to check that x1+c1(x2-x1) and x1+c2(x2-x1) are two different vectors. This shows that there are infinitely many solutions.