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).
Feb 27th 2007, 12:09 PM
Suppose we use x & y as two different solutions to AX=B.
Then A[x+c(y-x)]=Ax+cAy-cAx=B+cB-cB=B for every c.
Thus we have infinitely many solutions.
Feb 27th 2007, 12:49 PM
Thanks. Are there any particular theorems that proof those statements, though?