Math 2.0

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

Rouché–Capelli theorem

Builds on System of linear equationsAugmented matrixRank of a matrix

Statement

Let FF be a field, let AA be an m×nm \times n matrix over FF, and let b∈Fmb \in F^m. The system Ax=bAx = b has a solution if and only if rank⁡A=rank⁡[A∣b]\rank A = \rank [A \mid b], where [A∣b][A \mid b] is the augmented matrix and rank⁡\rank is the rank of a matrix. If a solution exists, then the solution set is an affine subspace of FnF^n of dimension n−rank⁡An - \rank A. In particular, the solution is unique if and only if rank⁡A=n\rank A = n.

Remarks

The theorem is also called the Kronecker–Capelli theorem. If rank⁡[A∣b]=rank⁡A+1\rank [A \mid b] = \rank A + 1, then the system is inconsistent. The dimension n−rank⁡An - \rank A is the number of free variables.

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Linear equation, definition, no evidence levelLinear equationdefinitionMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSystem of linear equations, definition, no evidence levelSystem of linear equa…definitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionCoefficient matrix, definition, no evidence levelCoefficient matrixdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionAugmented matrix, definition, no evidence levelAugmented matrixdefinitionColumn space, definition, no evidence levelColumn 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 matrixdefinitionRouché–Capelli theorem, statement, no evidence levelRouché–Capelli theoremstatement
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. Linear equation definition
  2. Matrix definition
  3. Vector space definition
  4. System of linear equations definition
  5. Linear combination definition
  6. Linear subspace definition
  7. Coefficient matrix definition
  8. Linear independence definition
  9. Linear span definition
  10. Augmented matrix definition
  11. Column space definition
  12. Spanning set definition
  13. Basis definition
  14. Finite-dimensional vector space definition
  15. Dimension definition
  16. Rank of a matrix definition
  17. Rouché–Capelli theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Rouché–Capelli_theorem, lead section
About this page
Tag0HV1SKWPMS
URIm2://0HV1SKWPMS
Aliaseskronecker-capelli-theorem
Kindstatement, theorem
WritingW1 stated
MSC15A06
StatusThis copy of the site has no build data. Run m2 status 0HV1SKWPMS.
UsesSystem of linear equations, Augmented matrix, Rank of a matrix
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