Overdetermined system

Remarks

Example: the system $x = 1$, $y = 2$, $x + y = 3$ over $\R$ has $m = 3$ equations and $n = 2$ unknowns, so it is overdetermined; it is consistent, with the solution $(x, y) = (1, 2)$. Non-example: the system $x + y = 1$, $x - y = 0$ has $m = n = 2$, so it is not overdetermined. If $m > n$, then the column space of $A$ has dimension at most $n < m$, so it is a proper subspace of $F^m$. Hence there are right-hand sides $b$ for which the system is inconsistent. Over $\R$, the consistent right-hand sides form a proper subspace of $\R^m$, so a randomly chosen $b$ almost never gives a consistent system. Over $\R$, the least squares solutions are the solutions of the normal equations $A^{T} A x = A^{T} b$, which are always consistent. A system with fewer equations than unknowns is called underdetermined.