Solution Types
Gaussian elimination reveals more than just the answer — it tells you whether a system has one solution, no solution, or infinitely many. The key is rank.
Every System Has One Fate
There is no fourth option. Every linear system Ax = b falls into exactly one of these three categories — and the RREF tells you which.
What Is Rank?
rank(A) = number of pivot positions = dimensions of information. Columns without pivots = free variables. Their count is n − rank(A).
The Consistency Test
A system Ax = b is consistent if and only if:
If b adds a new pivot — creating a row [0 0 … 0 | c] with c ≠ 0 — the system says "0 = c". Impossible. No solution.
The Contradiction Row
- RREF contains a row [0 0 … 0 | c], c ≠ 0
- This reads: 0 = c — a contradiction
- rank([A|b]) > rank(A)
- Geometrically: the planes never share a point
- b lies outside the column space of A
Full Rank
rank(A) = n. Every column has a pivot. Every variable is determined. A is invertible and x* = A⁻¹b.
Free Variables
rank(A) < n and consistent. Each non-pivot column gives a free variable. The solution set is a line, plane, or higher subspace.
Particular + Homogeneous
The full solution to Ax = b is always:
x_p: any particular solution (set free variables = 0). x_h: general solution to Ax = 0 (the null space). The null space has dimension n − rank(A).
Key Takeaways
- Three and only three solution types exist
- Consistency test: rank(A) = rank([A|b])
- Unique solution: rank(A) = n (full column rank)
- Infinitely many: consistent + rank(A) < n
- Free variables = n − rank(A)
- General solution = particular + null space
M3-L4: Applications
From theory to practice — see how linear systems power circuit analysis, network flow, data fitting, and more. Real problems, real matrices, real solutions.