Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Columns are independent iff every column has a pivot

Builds on Linear independencePivotMatrix

Statement

Let FF be a field and let AA be a matrix over FF. The columns of AA are linearly independent if and only if each column of AA contains a pivot position.

Remarks

Equivalently, the columns are dependent exactly when some column has no pivot. A column without a pivot gives a free variable of the system Ax=0Ax = 0, and then this system has a nonzero solution.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionRow echelon form, definition, no evidence levelRow echelon formdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionPivot, definition, no evidence levelPivotdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionColumns are independent iff every column has a pivot, statement, no evidence levelColumns are independe…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.
  • no evidence level
  1. Matrix definition
  2. Vector space definition
  3. Row echelon form definition
  4. Linear combination definition
  5. Pivot definition
  6. Linear independence definition
  7. Columns are independent iff every column has a pivot statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-4.xml#L739, src/section2-4.xml:739
About this page
Tag55K7JBE13G
URIm2://55K7JBE13G
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 55K7JBE13G.
UsesLinear independence, Pivot, Matrix
Changed4 November 2026

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