1. Limited time only! Sign up for a free 30min personal 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!

Definition of linearly independent

  1. May 9, 2010 #1
    I've looked it up and can't really find a clear answer..
    For a system of 3 simultaneous linear equations, is there any difference between 'the equations are linearly independent' and 'the equations have a unique solution'. If so what is it?
    Thanks!
     
  2. jcsd
  3. May 9, 2010 #2
    Two or more equations are linearly independent if they are:

    a) linear (ie, no terms with power > 1 and no multiplication of variables by each other),
    b) independent (none of the equations can be derived algebraically from the others).

    Some examples:

    Linearly independent:
    2x + 3y = 8
    3x + 2y = 5

    x + y + z = 7
    3x + 2y = 17

    Not linearly independent:
    x + y = 4
    2x + 2y = 8
    (second equation is a multiple of the first)

    x+y=3
    x+2y=9
    2x+3y=12
    (third equation is sum of the first two)

    x2 + y = 1
    x + 3y = 9
    (equation is quadratic, not linear)

    xy + 3y = 6
    (xy term makes equation not linear)

    A system of linearly independent equations need not be consistent, but if the left-hand sides of all the equations are linearly independent, then it always will be.
     
  4. May 9, 2010 #3
    Hmmn.. thanks that does help.
    I guess what I really meant is if a system of equations has a unique solution then is it always linearly independent?
     
  5. May 9, 2010 #4

    Mark44

    Staff: Mentor

  6. May 9, 2010 #5

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Condition (a) is not required. For example, the functions f(x)=x and g(x)=x2 are linearly independent.

    Condition (b) is not quite correct. Better said, a set of expressions {f1(x1,x2,...), f2(x1,x2,...), ...} are linearly independent if the only solution to a1f1+a2f2+...=0 is the trivial solution a1=a2=...=0.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Definition of linearly independent
  1. Linear independence (Replies: 6)

  2. Linear independance (Replies: 3)

  3. Linear independence (Replies: 3)

Loading...