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).
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 of x is often denoted
since it is given by
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
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, [0].
The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map.
Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. This gives one way in which to visualize quotient spaces geometrically.
Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1,…,xn). The subspace, identified with Rm, consists of all n-tuples such that the last n-m entries are zero: (x1,…,xm,0,0,…,0). Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last n−m coordinates. The quotient space Rn/ Rm is isomorphic to Rn−m in an obvious manner.
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