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Concept
Quotient space (linear algebra)
Summary
In linear algebra, the quotient of a vector space by a subspace is a vector space obtained by "collapsing" to zero. The space obtained is called a quotient space and is denoted (read " mod " or " by ").
Formally, the construction is as follows. Let be a vector space over a field , and let be a subspace of . We define an equivalence relation on by stating that if . That is, is related to if one can be obtained from the other by adding an element of . From this definition, one can deduce that any element of is related to the zero vector; more precisely, all the vectors in get mapped into the equivalence class of the zero vector.
The equivalence class – or, in this case, the coset – of is often denoted
since it is given by
The quotient space is then defined as , the set of all equivalence classes induced by on . Scalar multiplication and addition are defined on the equivalence classes by
for all , and
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representatives). These operations turn the quotient space into a vector space over with being the zero class, .
The mapping that associates to the equivalence class is known as the quotient map.
Alternatively phrased, the quotient space is the set of all affine subsets of which are parallel to .
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. Note that the points along any one such line will satisfy the equivalence relation because their difference vectors belong to Y. This gives a way to visualize quotient spaces geometrically. (By re-parameterising these lines, the quotient space can more conventionally be represented as the space of all points along a line through the origin that is not parallel to Y.
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