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2.4 Homogeneous Systems 63
2.4.5. Suppose that A is the coefficient matrix for a homogeneous system of
four equations in six unknowns and suppose that A has at least one
nonzero row.
(a) Determine the fewest number of free variables that are possible.
(b) Determine the maximum number of free variables that are pos-
sible.
2.4.6. Explain why a homogeneous system of m equations in n unknowns
where m<n must always possess an infinite number of solutions.
2.4.7. Construct a homogeneous system of three equations in four unknowns
that has
−2 −3
1 0
0 + x 4 2
x 2
0 1
as its general solution.
2.4.8. If c 1 and c 2 are columns that represent two particular solutions of
the same homogeneous system, explain why the sum c 1 + c 2 must also
represent a solution of this system.
