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Free variable

  1. Jan 31, 2014 #1
    1. The problem statement, all variables and given/known data
    |1 0-1-2-8 | -3|
    |0 0 1 2 7 | 1|
    |0 0 0 1 -4 | -13|


    I think that x[itex]_{2}[/itex] is a free variable, so when writing the general solution in a solution set, what would I put for it? Nothing?
     
  2. jcsd
  3. Jan 31, 2014 #2

    Mark44

    Staff: Mentor

    You could say that x2 is arbitrary or you could say that x2 = t, an arbitrary real number.
     
  4. Jan 31, 2014 #3
    Oh ok thanks for that, I got it.
     
  5. Jan 31, 2014 #4

    Ray Vickson

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    When you write out the equations in detail you see that there is no ##x_2## anywhere in the system. I would not put anything for it; it does not "exist" for this system, no more than ##x_{17}## or ##x_{265}## exist here. However, I suppose you *could* argue the point.
     
  6. Jan 31, 2014 #5

    Dick

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    I would say ##x^2+y^2=1## in ##R^3## represents a cylinder. Saying it's a circle in the plane because z doesn't occur isn't really a good answer.
     
  7. Jan 31, 2014 #6

    Ray Vickson

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    I agree, but in the case herein, the "appropriate" view depends on what space you want to operate in.

    In ##\mathbb{R}^5## the component ##x_2## is arbitrary, so the described "figure" is like a cylinder parallel to the 2-axis. However, we can look at the problem instead in ##\mathbb{R}^3##, where the problem is to represent the (column) vector ##(-3,1,-13)^T## as a linear combination of the 3-dimensional vectors in columns 1--5. The second column is the zero vector, so would have no effect at all on anything in the problem. Having an extraneous ##x_2## does not change the geometry in any way.
     
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