The underdetermined case, by contrast, occurs when the system has been underconstrained—that is, when the unknowns outnumber the equations.
There are an infinity of such solutions, which form a vector space, whose dimension is the difference between the number of unknowns and the rank of the matrix of the system.
If an underdetermined system of t equations in n variables (t < n) has solutions, then the set of all complex solutions is an algebraic set of dimension at least n - t. If the underdetermined system is chosen at random the dimension is equal to n - t with probability one.
In general, an underdetermined system of linear equations has an infinite number of solutions, if any.
However, in optimization problems that are subject to linear equality constraints, only one of the solutions is relevant, namely the one giving the highest or lowest value of an objective function.