54. Linear System Solutions
A linear system has exactly one of three outcomes:
- Unique solution — exactly one satisfies the system
- No solution — the system is inconsistent
- Infinitely many solutions — the system is underdetermined
The REF (or RREF) of the augmented matrix tells you which case you’re in.
54.1. 1. Unique solution
Every column of has a pivot, no free variables, and no inconsistency:
Equivalently: (the number of unknowns).
54.2. 2. No solution (inconsistent)
A row of the form with — translates to “” which is impossible.
The second row says . No satisfies this.
Equivalently: .
54.3. 3. Infinitely many solutions (free variables)
Some columns of have no pivot — those correspond to free variables you can set arbitrarily; the pivot variables are determined by them.
Columns 2 and 4 have no pivot → are free.
Equivalently: . The solution set is an affine subspace of dimension — see Rank–Nullity.
54.4. Rouché–Capelli summary
| Case | Rank condition | Solution set |
|---|---|---|
| Unique | single point | |
| None | empty | |
| Infinite | affine subspace of dim |
54.5. See also
- Row Echelon Form — the algorithm
- Linear System Special Cases — when zero rows appear
- Rank and Rank–Nullity
- Homogeneous System — special case