Definition
Formally, the construction is as follows (Halmos 1974, §21-22). Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector; in other words all the vectors in N get mapped into the equivalence class of the zero vector.
The equivalence class of x is often denoted
- = x + N
since it is given by
- = {x + n : n ∈ N}.
The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplication and addition are defined on the equivalence classes by
- α = for all α ∈ K, and
- + = .
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space V/N into a vector space over K with N being the zero class, .
The mapping that associates to v ∈ V the equivalence class is known as the quotient map.
Read more about this topic: Quotient Space (linear Algebra)
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