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In describing the solution of a system of linear equations with many solutions, why do we use a free variable as a parameter to describe the other variables in the solution? Why do we not we use a leading variable? Since by the commutative property of addition we can swap between the free and leading variables, e.g. x + y + z = x + z + y; the solution set will essentially be the same (albeit having different orders).

Definitions:

  • A free variable is a parameter that is not a leading variable.
  • A leading variable is the first variable that has a non-zero coefficient in reduced form.
  • These definitions are most easily understood with respect to the Echelon form of a system of linear equations expressed as a Matrix. See this link for details.

For example:

Let $S$ be the solution set of the system $$\begin{align*} x+y+z &= 3\\ y-z &= 4 \end{align*}$$

Using the free variable $z$ as the parameter $$S = \{(-2z-1, z+4, z)\mid z\in\mathbb{R}\}.$$

Using the leading variable $y$ as the parameter $$S = \{(-2y+7, y, y-4)\mid y\in\mathbb{R}\}.$$

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    Well, it all seems to depend on the way you order your variables (in more general context choose your basis) - the definitions are order (basis) specific, but the equations aren't. So the distinction is an artificial one, no doubt useful in the applications being considered in your source (you have to begin somewhere, let's choose the leading variable to eliminate).2012-03-25
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    @MarkBennet: Thanks for the quick comment! I was wondering if there is perhaps something with the leading variable in the succeeding development of the theory which makes it special or something. (I have only barely passed the first chapter of LA.)2012-03-25

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