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Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables
Statement
Let be the RREF of . The system is consistent exactly when the last column of is not a pivot column. When it is consistent, every nonpivot variable may be chosen arbitrarily, and each pivot variable is then determined uniquely by its row. This parametrises every solution exactly once.
Facts & Assumptions
Given: A finite system over a field and the RREF of its augmented matrix.
Row operations on an augmented matrix preserve exactly its solution set (Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system).
Every finite matrix has a unique RREF (Every finite matrix over a field is row equivalent to exactly one reduced row echelon matrix).
The augmented matrix records the coefficients and right-hand side of (Matrix equation , its solution set, consistency, homogeneous systems and the augmented matrix ).
In RREF, every pivot is the only nonzero entry in its column (Row echelon form, reduced row echelon form, leading entries, pivots, pivot positions and pivot columns).
Proof
Replace by its unique RREF . By [L1] this changes no solution.
A pivot in the augmented column produces a row , after its pivot has been normalized, so the system is inconsistent. Conversely, if there is no such pivot, every nonzero row has a coefficient pivot and imposes an equation for that pivot variable.
Choose arbitrary values for the nonpivot variables. Each pivot row then determines its pivot variable uniquely, and substituting these values satisfies every nonzero row while zero rows impose nothing. Every solution has those same free coordinates and therefore arises uniquely from this construction.
Depends on
- Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system
- Every finite matrix over a field is row equivalent to exactly one reduced row echelon matrix
- Matrix equation $Ax=b$, its solution set, consistency, homogeneous systems and the augmented matrix $[A\mid b]$
- Row echelon form, reduced row echelon form, leading entries, pivots, pivot positions and pivot columns
Used by
- Over an infinite field, a finite linear system has no solution, exactly one solution, or infinitely many solutions according to its pivots Corollary
- The system Ax=b is consistent if and only if rank(A)=rank([A∣ b]) Corollary
- A 3×3 system has no solution because its augmented column contains a pivot Example
- A 3×3 system row reduces to a unique solution Example
- A consistent underdetermined system has an affine two-parameter solution set Example
- The rank and solution behaviour of a parameterised matrix change at one exceptional parameter Example
- Invertible matrix theorem: invertibility, full pivot rank, RREF I, trivial nullspace and unique solvability are equivalent Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 25 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- D. Margalit and J. Rabinoff, Interactive Linear Algebra, §§1.3 and 2.4 (standard reference, not scraped)