Proof of a system of linear equations

[arpitm08]

Proof of a system of linear equations!

Homework Statement

Prove that if more than one solution to 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 all these solutions are different.)

Homework Equations

none that i know of

The Attempt at a Solution

This is what i have so far,

Let x1 and x2 be different solutions to Ax=B...

I don't know where to go from there. How do i show that x1 + c(x2 - x1) is also a solution. Should i create an m by n matrix and then show that it works? And then how would I show that these solutions are different? Please help. Thanks in advance.

 

[LCKurtz]

"Let x1 and x2 be different solutions to Ax=B"

so Ax1 = B and Ax2 = B, right?

How would you check whether x1 + c(x2-x1) is a solution?

 

[arpitm08]

Could you do this..
A[(1-c)x1 + c x2] = (1-c)Ax1 + c Ax2 = (1-c) B + c B = B.
Wow that works! I knew I was really close. Hahah.
How do I show that these are all different?

Would I have to show that x1 or x2 is not equal to x1 + c(x2 - x1)??

 

[Dick]

Could you do this..
A[(1-c)x1 + c x2] = (1-c)Ax1 + c Ax2 = (1-c) B + c B = B.
Wow that works! I knew I was really close. Hahah.
How do I show that these are all different?

Would I have to show that x1 or x2 is not equal to x1 + c(x2 - x1)??

No, you have to show that if c1 and c2 are different then x1+c1(x2-x1) and x1+c2(x2-x1) are different. That would show there were an infinite number of solutions. Isn't that what the question says?

 

[arpitm08]

Yea, Thanks!