Math 2.0

definition

Rank of a linear map

Definition

Let FF be a field, let VV and WW be vector spaces over FF, and let T ⁣:V→WT \colon V \to W be a linear map. The rank of TT is rank⁡T=dim⁡im⁡T,\rank T = \dim \im T, the dimension of the image of TT.

Remarks

For the map x↦Axx \mapsto Ax of a matrix AA, the rank is the rank of the matrix. The rank satisfies rank⁡T≤dim⁡V\rank T \le \dim V and rank⁡T≤dim⁡W\rank T \le \dim W. The rank–nullity theorem relates the rank and the nullity. The rank of a composition satisfies rank⁡(T∘S)≤min⁡(rank⁡T,rank⁡S)\rank(T \circ S) \le \min(\rank T, \rank S).

Explanations

L1 human

The rank of a linear map counts the dimensions of its output. A map from space onto a plane has rank 22. The zero map has rank 00.

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To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionImage of a linear map, definition, no evidence levelImage of a linear mapdefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning 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 linear map, definition, no evidence levelRank of a linear mapdefinition
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Vector space definition
  2. Linear combination definition
  3. Linear map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Image of a linear map definition
  7. Linear span definition
  8. Spanning set definition
  9. Basis definition
  10. Finite-dimensional vector space definition
  11. Dimension definition
  12. Rank of a linear map definition

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2 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Rank_(linear_algebra), lead section

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