A system that is underdetermined and homogeneous will typically have what characteristic?

An underdetermined system refers to a scenario where there are fewer equations than unknowns. In the context of a homogeneous system, which is defined as having all constant terms equal to zero, this system will typically exhibit certain characteristics due to its structure.

For a homogeneous system to have infinitely many solutions, we can consider the fact that the system can be represented in the form Ax = 0, where A is the coefficient matrix. Since there are more variables than equations, the rank of the matrix A (the maximum number of linearly independent rows) is less than the number of variables. This results in at least one free variable, which allows for a solution space that is spanned by vectors.

Because of this free variable, the solutions can take on an infinite number of values, meaning that the system has infinitely many solutions. This property is intrinsic to underdetermined homogeneous systems, especially when the homogeneous case (where the constants are zero) allows for a solution vector that is not restricted to a single outcome.

Considering other possibilities, a unique solution would require a square system (equal number of equations and unknowns) and full rank. There cannot be no solution in a homogeneous system because it will always at least satisfy the trivial solution (all zero