Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Uniqueness of solutions of a consistent linear system by pivot columns

Builds on Consistent and inconsistent systemsSystem of linear equationsPivot

Statement

Take a system of linear equations over the real numbers that is consistent. Look at the pivot positions of its coefficient matrix.

If each column of the coefficient matrix holds a pivot position, then the system has exactly one solution.

If at least one column of the coefficient matrix holds no pivot position, then the system has infinitely many solutions.

Remarks

A column with a pivot position gives a basic variable. A column without one gives a free variable. So the two cases say: no free variable means one solution, and at least one free variable means infinitely many solutions.

Consistency is needed. An inconsistent system has no solution, whatever its pivot columns are.

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Consistent and inconsistent systems, concept, no evidence levelConsistent and incons…conceptMatrix, definition, no evidence levelMatrixdefinitionSystem of linear equations, concept, no evidence levelSystem of linear equa…conceptRow echelon form, definition, no evidence levelRow echelon formdefinitionPivot, definition, no evidence levelPivotdefinitionUniqueness of solutions of a consistent linear system by pivot columns, statement, no evidence levelUniqueness of solutio…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Consistent and inconsistent systems concept
  2. Matrix definition
  3. System of linear equations concept
  4. Row echelon form definition
  5. Pivot definition
  6. Uniqueness of solutions of a consistent linear system by pivot columns statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section1-4.xml#L917, src/section1-4.xml:917
About this page
Tag5MGKEK1K42
URIm2://5MGKEK1K42
Kindstatement, theorem
WritingW0 stub
MSC15A06
StatusThis copy of the site has no build data. Run m2 status 5MGKEK1K42.
UsesConsistent and inconsistent systems, System of linear equations, Pivot
Changed4 November 2026

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