Math 2.0

Linear and multilinear algebra; matrix theory Definition

Quotient map

Builds on Quotient space

Definition

Let FF be a field, let VV be a vector space over FF, and let WW be a subspace of VV. The quotient map is the map π ⁣:V→V/W,π(v)=v+W,\pi \colon V \to V / W, \qquad \pi(v) = v + W, from VV to the quotient space V/WV / W.

Remarks

The map π\pi is linear and surjective, and its kernel is WW. Each linear map T ⁣:V→UT \colon V \to U with W⊆ker⁡TW \subseteq \ker T factors through π\pi, see universal property of the quotient space. In Mathlib, it is Submodule.mkQ.

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Vector space, definition, no evidence levelVector spacedefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionQuotient space, definition, no evidence levelQuotient spacedefinitionQuotient map, definition, no evidence levelQuotient mapdefinition
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  1. Vector space definition
  2. Linear subspace definition
  3. Quotient space definition
  4. Quotient map definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra), Definition
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