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Homework Help: 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?
  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)

    (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


    Staff: Mentor

  6. May 9, 2010 #5

    D H

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    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.
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