Math 2.0

Linear and multilinear algebra; matrix theory Definition

Nullity

Builds on Kernel of a linear mapDimension

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 nullity of TT is nullity⁡T=dim⁡ker⁡T,\operatorname{nullity} T = \dim \ker T, the dimension of the kernel of TT. The nullity of a matrix AA is the nullity of the map x↦Axx \mapsto Ax, the dimension of the null space of AA.

Remarks

The nullity is 00 if and only if TT is injective. For an m×nm \times n matrix, the nullity is the number of free variables of Ax=0Ax = 0. By the rank–nullity theorem, rank⁡T+nullity⁡T=dim⁡V\rank T + \operatorname{nullity} T = \dim V when VV is finite-dimensional.

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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 independencedefinitionKernel 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 levelDimensiondefinitionNullity, definition, no evidence levelNullitydefinition
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. Vector space definition
  2. Linear combination definition
  3. Linear map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Kernel 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. Nullity definition

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2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Kernel_(linear_algebra), Properties
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MSC15A03, 15A04
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