Page 103 - Advanced Linear Algebra
P. 103
Chapter 3
The Isomorphism Theorems
Quotient Spaces
Let be a subspace of a vector space . It is easy to see that the binary relation
:
=
on defined by
=
"# ¯ " c #:
#
is an equivalence relation. When " # , we say that and are congruent
"
modulo . The term mod is used as a colloquialism for modulo and " # is
:
often written
"# mod :
When the subspace in question is clear, we will simply write "# .
To see what the equivalence classes look like, observe that
´#µ~¸"= "#¹
~¸"= " c #:¹
~¸"= "~# b for some :¹
~¸# b :¹
~# b :
The set
´#µ~# b : ~¸# b :¹
is called a coset of in and is called a coset representative for # b . :
:
#
=
(Thus, any member of a coset is a coset representative. )
The set of all cosets of in is denoted by
=
:
=°: ~ ¸# b : # =¹
This is read “ mod ” and is called the quotient space of modulo : . Of
=
=
:
