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

• Mar 25th 2009, 12:07 PM
paulrb
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.
• Mar 25th 2009, 12:15 PM
o_O
Consider: $\displaystyle A \left[X_1 + c\left(X_2 - X_1\right)\right]$

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

$\displaystyle = AX_1 + cAX_2 - cAX_1$

$\displaystyle = \cdots$

$\displaystyle = B$

___________

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