Math 2.0

definition

Quotient map

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.

Explanations

L1 human

The quotient map sends each vector to the class of all vectors that differ from it by an element of WW. It forgets exactly the information in WW. The vectors that it sends to zero are exactly the vectors of WW.

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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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Sources

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

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