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Quotient Vector Space


Suppose that V={(x_1,x_2,x_3)} and W={(x_1,0,0)}. Then the quotient space V/W (read as "V mod W") is isomorphic to {(x_2,x_3)}=R^2.

In general, when W is a subspace of a vector space V, the quotient space V/W is the set of equivalence classes [v] where v_1∼v_2 if v_1-v_2 in W. By "v_1 is equivalent to v_2 modulo W," it is meant that v_1=v_2+w for some w in W, and is another way to say v_1∼v_2. In particular, the elements of W represent [0]. Sometimes the equivalence classes [v] are written as cosets v+W.

The quotient space is an abstract vector space, not necessarily isomorphic to a subspace of V. However, if V has an inner product, then V/W is isomorphic to

 W^_|_={v:<v,w>=0 for all w in W}.

In the example above, W^_|_={(0,x_2,x_3)}.

Unfortunately, a different choice of inner product can change W^_|_. Also, in the infinite-dimensional case, it is necessary for W to be a closed subspace to realize the isomorphism between V/W and W^_|_, as well as to ensure the quotient space is a T2-space.


See also

Coset, Orthogonal Set, Quotient Space, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Quotient Vector Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/QuotientVectorSpace.html

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