Math 2.0

Linear and multilinear algebra; matrix theory Systems of linear equations Theorem

Row rank equals column rank

Builds on Rank of a matrixRow spaceTranspose

Statement

Let FF be a field and let AA be an m×nm \times n matrix over FF. Then the dimension of the row space of AA equals the dimension of the column space of AA. Equivalently, rank⁡AT=rank⁡A\rank A^T = \rank A, where ATA^T is the transpose.

Remarks

One proof uses row reduction: row operations keep the row space and the dimension of the column space. In reduced row echelon form, both dimensions equal the number of pivots. Another proof uses the annihilator: the dual map has the transposed matrix and the same rank. Over ℂ\C, also rank⁡A∗=rank⁡A\rank A^* = \rank A.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionTranspose, definition, no evidence levelTransposedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionColumn space, definition, no evidence levelColumn spacedefinitionRow space, definition, no evidence levelRow spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionRank of a matrix, definition, no evidence levelRank of a matrixdefinitionRow rank equals column rank, statement, no evidence levelRow rank equals colum…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. Transpose definition
  4. Linear combination definition
  5. Linear subspace definition
  6. Linear independence definition
  7. Linear span definition
  8. Column space definition
  9. Row space definition
  10. Spanning set definition
  11. Basis definition
  12. Finite-dimensional vector space definition
  13. Dimension definition
  14. Rank of a matrix definition
  15. Row rank equals column rank statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Rank_(linear_algebra), Proofs that column rank = row rank
About this page
Tag0TZAG60ZYT
URIm2://0TZAG60ZYT
Kindstatement, theorem
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 0TZAG60ZYT.
UsesRank of a matrix, Row space, Transpose

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