1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof of a system of linear equations

  1. Sep 21, 2009 #1
    Proof of a system of linear equations!!

    1. The problem statement, all variables and given/known data

    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.)

    2. Relevant equations

    none that i know of

    3. 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.
     
  2. jcsd
  3. Sep 21, 2009 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Proof of a system of linear equations!!

    "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?
     
  4. Sep 21, 2009 #3
    Re: Proof of a system of linear equations!!

    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)??
     
  5. Sep 21, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Proof of a system of linear equations!!

    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?
     
    Last edited: Sep 21, 2009
  6. Sep 21, 2009 #5
    Re: Proof of a system of linear equations!!

    Yea, Thanks!!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof of a system of linear equations
Loading...