Quotient Vector Space
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المرجع الالكتروني للمعلوماتيه
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www.almerja.com
الجزء والصفحة:
...
4-8-2021
1890
Quotient Vector Space
Suppose that ![V=<span style=]()
{(x_1,x_2,x_3)}" src="https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/Inline1.gif" style="height:15px; width:100px" /> and ![W=<span style=]()
{(x_1,0,0)}" src="https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/Inline2.gif" style="height:15px; width:91px" />. Then the quotient space 
(read as "
mod 
") is isomorphic to ![<span style=]()
{(x_2,x_3)}=R^2" src="https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/Inline6.gif" style="height:17px; width:85px" />.
In general, when 
is a subspace of a vector space 
, the quotient space 
is the set of equivalence classes ![[v]](https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/Inline10.gif)
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 ![[0]](https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/Inline21.gif)
. Sometimes the equivalence classes ![[v]](https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/Inline22.gif)
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^_|_=<span style=]() {v:<v,w>=0 for all w in W}. " src="https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/NumberedEquation1.gif" style="height:15px; width:201px" /> |
In the example above, ![W^_|_=<span style=]()
{(0,x_2,x_3)}" src="https://mathworld.wolfram.com/images/equations/QuotientVectorSpace/Inline27.gif" style="height:15px; width:105px" />.
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.
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