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- Apr 14, 2013

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Let an arbitrary linear system of $3$ equations and $3$ variables be given. There are $4$ cases how the planes can be related.

Describe these $4$ cases graphically and describe the set of solutions in each case.

I have done the following:

If the three equations are linearly independent, then the system has a single solution.

In this case the three planes described by the equations intersect, since they are neither parallel nor identical.

The intersection of these three planes is a point in space.

If two of the three equations are linearly independent, then the system has a set of solution with one free parameter.

In this case the two independent planes described by the equations intersect, since they are neither parallel nor identical. The third one is either parallel or identical to one of the other ones.

The intersection of these two planes is a line in space.

If all the three equations are linearly dependent, then the system has either a set of solution with two free parameters or no solution (empty set of solutions).

In this case the three planes described by the equations are either identical or parallel.

The intersection of these three planes is either a plane (if the planes are identical) or there is no intersection (if the planes are parallel).

Are the four cases correct and complete? Could we improve something?