Ax = 0 is a homogeneous system of linear equations. 0 represents the zero vector. Every homogeneous system of linear equations has at least one solution which is the trivial solution or in other words the zero vector. A homogeneous system of linear equations with a nontrivial solution can be thought of as a set of vectors that when summed together start at the origin and end at the origin. Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.
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