Math 2.0

statement theorem

Rank–nullity theorem

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let WW be a vector space over FF. Let T ⁣:V→WT \colon V \to W be a linear map. Then dim⁡im⁡T+dim⁡ker⁡T=dim⁡V,\dim \im T + \dim \ker T = \dim V, that is, rank⁡T+nullity⁡T=dim⁡V\rank T + \operatorname{nullity} T = \dim V, with the rank and the nullity of TT.

Remarks

For an m×nm \times n matrix AA, this says rank⁡A+dim⁡{x:Ax=0}=n\rank A + \dim \{x : Ax = 0\} = n. A proof extends a basis of ker⁡T\ker T to a basis of VV; the images of the new vectors form a basis of im⁡T\im T. The theorem also follows from the first isomorphism theorem V/ker⁡T≅im⁡TV / \ker T \cong \im T.

Explanations

L1 human

Each dimension of the domain either survives in the image or is crushed into the kernel. So the dimensions of image and kernel add up to the dimension of the domain. A map from R5\R^5 to R3\R^3 has a kernel of dimension at least 22.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

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What this needs

To read this node, read these 15 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 mapdefinitionKernel of a linear map, definition, no evidence levelKernel 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 mapdefinitionNullity, definition, no evidence levelNullitydefinitionRank–nullity theorem, statement, no evidence levelRank–nullity theoremstatement
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. Kernel of a linear map definition
  8. Linear span definition
  9. Spanning set definition
  10. Basis definition
  11. Finite-dimensional vector space definition
  12. Dimension definition
  13. Rank of a linear map definition
  14. Nullity definition
  15. Rank–nullity theorem statement

What depends on this

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Sources

  1. Web https://en.wikipedia.org/wiki/Rank–nullity_theorem, Stating the theorem

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