To express solutions in parametric vector form, which equation must be solved?

To express solutions in parametric vector form, the equation that is relevant is the homogeneous equation, which is represented as Ax = 0. This equation is fundamental in linear algebra because solving it allows us to identify the solution space of the system of equations represented by the matrix A.

When you solve Ax = 0, you are looking for the null space of the matrix A. This null space includes all vectors x that, when multiplied by A, yield the zero vector. The solutions to this equation can often be expressed in parametric form, where free variables correspond to dimensions of the solution space. Each free variable can be associated with a parameter that describes a line or a higher-dimensional surface in the vector space.

In contrast, the other equations, such as Ax = b for some vector b (which represents a non-homogeneous system), do not yield the generic parametric vector form that encompasses all solutions, as they typically produce a single solution or no solution at all, depending on the relationship between A and b. Solving Ax = 1 or Ax = 2 similarly constrains the solutions to specific scenarios that do not generally allow for a full parametric description of the solution set.

In summary, the homogeneous equation Ax = 0