Math 2.0

Linear and multilinear algebra; matrix theory Definition

Rank of a matrix

Builds on Column spaceDimension

Definition

Let FF be a field and let AA be an m×nm \times n matrix over FF. The rank of AA is rank⁡A=dim⁡Col⁡(A),\rank A = \dim \operatorname{Col}(A), the dimension of the column space of AA.

Remarks

The rank is the maximal number of linearly independent columns of AA. It also equals the dimension of the row space, by row rank equals column rank. It equals the number of pivots in a row echelon form of AA, and the largest order of a nonzero minor of AA. An n×nn \times n matrix is invertible if and only if its rank is nn.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionLinear 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 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 matrixdefinition
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. Matrix definition
  2. Vector space definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear span definition
  7. Column space definition
  8. Spanning set definition
  9. Basis definition
  10. Finite-dimensional vector space definition
  11. Dimension definition
  12. Rank of a matrix definition

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6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Rank_(linear_algebra), lead section
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WritingW1 stated
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DefinesRank of a matrix
UsesColumn space, Dimension

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