Editing Quotient space (linear algebra) (section)

== Definition ==
Formally, the construction is as follows.<ref>{{Harvard citation text|Halmos|1974}} pp. 33-34 §§ 21-22</ref> Let <math>V</math> be a [[vector space]] over a [[field (mathematics)|field]] <math>\mathbb{K}</math>, and let <math>N</math> be a [[linear subspace|subspace]] of <math>V</math>. We define an [[equivalence relation]] <math>\sim</math> on <math>V</math> by stating that <math>x \sim y</math> iff {{nowrap| <math>x - y \in N</math>}}. That is, <math>x</math> is related to <math>y</math> if and only if one can be obtained from the other by adding an element of <math>N</math>. This definition implies that any element of <math>N</math> is related to the zero vector; more precisely, all the vectors in <math>N</math> get mapped into the [[equivalence class]] of the zero vector.

The equivalence class – or, in this case, the [[coset]] – of <math>x</math> is defined as 
:<math>[x] := \{ x + n: n \in N \}</math>
and is often denoted using the shorthand <math>[x] = x + N</math>.

The quotient space <math>V/N</math> is then defined as <math>V/_\sim</math>, the set of all equivalence classes induced by <math>\sim</math> on <math>V</math>. Scalar multiplication and addition are defined on the equivalence classes by<ref>{{Harvard citation text|Katznelson|Katznelson|2008}} p. 9 § 1.2.4</ref><ref>{{Harvard citation text|Roman|2005}} p. 75-76, ch. 3</ref> 
*<math>\alpha [x] = [\alpha x]</math> for all <math>\alpha \in \mathbb{K}</math>, and
*<math>[x] + [y] = [x+y]</math>.
It is not hard to check that these operations are [[well-defined]] (i.e. do not depend on the choice of [[representative (mathematics)|representatives]]). These operations turn the quotient space <math>V/N</math> into a vector space over <math>\mathbb{K}</math> with <math>N</math> being the zero class, <math>[0]</math>.

The mapping that associates to {{nowrap|<math>v \in V</math>}} the equivalence class <math>[v]</math> is known as the '''quotient map'''.

Alternatively phrased, the quotient space <math>V/N</math> is the set of all [[Affine space|affine subsets]] of <math>V</math> which are [[Parallel (geometry)|parallel]] to {{nowrap|<math>N</math>.}}<ref>{{Harvard citation text|Axler|2015}} p. 95, § 3.83</ref>