Page 104 - Advanced Linear Algebra

88 Advanced Linear Algebra

course, the term space is a hint that we intend to define vector space operations
on =°: .

The natural choice for these vector space operations is

²" b:³b²#b:³ ~ ²"b#³b:
and
²" b :³ ~ ² "³ b :

but we must check that these operations are well-defined, that is,

1) " b :~ " b :Á # b :~ # b :¬ ²" b # ³ b :~ ²" b # ³ b :

2) "b : ~ "b : ¬ "b : ~ "b :

Equivalently, the equivalence relation – must be consistent with the vector
space operations on , that is,
=
3) "– " Á #– #¬ ²" b # ³ – ²" b # ³

4) "– "¬ "– "

This senario is a recurring one in algebra. An equivalence relation on an
algebraic structure, such as a group, ring, module or vector space is called a
congruence relation if it preserves the algebraic operations. In the case of a
vector space, these are conditions 3) and 4) above.
: " – "
These conditions follow easily from the fact that is a subspace, for if
, then
and #– #
"c "  :Á #c #  : ¬ ²"c " ³ b ²# c # ³  :

¬ ² " b # ³ c² " b # ³  :

¬ "b # – "b #
which verifies both conditions at once. We leave it to the reader to verify that
=°: is indeed a vector space over under these well-defined operations.
-
Actually, we are lucky here: For any subspace of , the quotient = ° : is a
=
:
vector space under the natural operations. In the case of groups, not all
subgroups have this property. Indeed, it is precisely the normal subgroups of
5
. . that have the property that the quotient ° 5 is a group. Also, for rings, it is
precisely the ideals (not the subrings) that have the property that the quotient is
a ring.
Let us summarize.