Convert equations into structure, then classify every possible solution set
Gaussian elimination replaces a system with an equivalent system that is easier to read. Row operations preserve the solution set while exposing pivots, free variables, contradictions, rank, and the geometry of the solution.
From equations to Ax=b
Separate coefficients, unknowns, and constants
Equation form
2x − y + z = 3
−x + y + 2z = 2
Matrix form
| 1 | 2 | −1 |
| 2 | −1 | 1 |
| −1 | 1 | 2 |
| x |
| y |
| z |
| 3 |
| 3 |
| 2 |
The augmented matrix appends b as a final column: [A|b]. Row operations act on entire equations, including the constant column.
Elementary row operations
Change representation without changing the solution set
Swap rows
Reorder equations. This is useful when a better pivot appears lower in the matrix.
Scale a row
Multiply an equation by a nonzero scalar. Never scale by zero.
Replace a row
Add a multiple of one row to another to eliminate an entry.
Column operations are not equation operations
Ordinary Gaussian elimination uses row operations because rows represent equations. Arbitrary column operations change the variables and therefore change the interpretation unless that change is explicitly tracked.
Pivots, free variables, and rank
Read the geometry from echelon structure
Pivot variables
A leading nonzero entry identifies a pivot column. The associated variable is determined by the equations once free variables are selected.
Free variables
A nonpivot variable can be assigned a parameter. Each free variable contributes one dimension to the null space.
| RREF evidence | Conclusion | Geometry |
|---|---|---|
| Pivot in every variable column; no contradiction | Unique solution | A single point. |
| At least one free variable; no contradiction | Infinitely many solutions | A line, plane, or higher-dimensional affine subspace. |
| Row [0 … 0 | c] with c≠0 | No solution | The constraints do not intersect. |
Interactive RREF laboratory
Reduce systems and inspect every row operation
Gaussian elimination and RREF analyzer
Select the number of equations and variables. The final column is the right-hand side b.
Numerical perspective
Distinguish mathematical equivalence from stable computation
Exact classroom elimination often uses small integers and fractions. Numerical software instead chooses pivots to limit roundoff growth, commonly using partial pivoting. Solving Ax=b by elimination or a factorization is generally preferable to computing A⁻¹ explicitly and then multiplying by b.
Factorization viewpoint
Elimination can be organized as an LU factorization, A=LU (with row permutations when needed). One factorization can then solve many systems with the same A but different right-hand sides.
Review questions
Classify systems and interpret pivot structure