Over $\mathbb R$ or $\mathbb C$, a linear system has zero, one, or infinitely many solutions. An inconsistent system has none. For a consistent rank-$r$ system with $n$ unknowns, the solution set is $x_p+\ker A$ and has $n-r$ free parameters.
If $n-r=0$, the solution is unique. If it is positive, the infinite field supplies infinitely many distinct parameter choices. Equivalently, two distinct solutions generate an entire affine line of solutions.
Over a finite field $\mathbb F_q$, the corresponding number is $q^{n-r}$ when consistent. Thus the infinite-solution classification must not be transferred unchanged to coding theory.
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- A real or complex linear system has zero, one, or infinitely many solutionsFor systems over an infinite field , the possibilities are no solution, a unique solution, or infinitely many...
- A consistent system with more variables than equations has infinitely many solutions.Theorem (over an infinite field, such as the reals or complexes): If a consistent linear system has more variables...
- A homogeneous system with more variables than equations has infinitely many solutions.Theorem (over an infinite field, such as the reals or complexes): A homogeneous system $Ax = 0$ with more variables...
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