The number of solutions to a linear system
Theorem: A linear system has either:
- 0 solutions (inconsistent)
- 1 solution (unique)
- Infinitely many solutions
No other number of solutions is possible.
Why: If a system has two distinct solutions $x_1$ and $x_2$, then the line connecting them contains infinitely many solutions. For homogeneous systems, any point on the line through $x_1$ and $x_2$ is also a solution.
How to determine which case:
- Row reduce the augmented matrix $[A | b]$ to echelon form
- If there is a row $[0 \ \cdots \ 0 \ | \ b]$ with $b \neq 0$ → 0 solutions
- If consistent and no free variables → 1 solution
- If consistent and at least one free variable → infinitely many solutions