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November 3rd, 2014, 09:29 AM   #1
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Systems of Linear Equations(Immediate)

Suppose that a system of linear equation in four variables has ten equations. Does it mean that the
solution set cannot have free variables?
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November 3rd, 2014, 09:36 AM   #2
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It depends.

If the equations are inconsistent, there are zero solutions.

If the equations are consistent and there is a set of four of them that are linearly independant, there will be a unique solution with no free variables.

If the equations are consistent and there is a set of three of them that are linearly independant (but no such set of four), there will be a solution having one free variable.

If the equations are consistent and there is a set of two of them that are linearly independant (but no such set of three or four), there will be a solution with two free variables.

If the equations are consistent but none are linearly independant (i.e. all the equations are equivalent), there will be a solution having three free variables.
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