Quotient Vector Space

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

In general, when is a subspace of a vector space , the quotient space is the set of equivalence classes where if . By " is equivalent to modulo ," it is meant that for some in , and is another way to say . In particular, the elements of represent . Sometimes the equivalence classes are written as cosets .

The quotient space is an abstract vector space, not necessarily isomorphic to a subspace of . However, if has an inner product, then 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 . Also, in the infinite-dimensional case, it is necessary for to be a closed subspace to realize the isomorphism between and , as well as to ensure the quotient space is a T2-space.

Quotient Vector Space

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