Quotient space (linear algebra)

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by 'collapsing' N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N). In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by 'collapsing' N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N). 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; more precisely all the vectors in N get mapped into the equivalence class of the zero vector. The equivalence class (or, in this case, the coset) of x is often denoted