Graphics Reference
In-Depth Information
(
)
=+
c
vW
+
c
vW
for each coset
v
+
W
in
V
/
W
and c Πk.
Definition.
The vector space
V
/
W
is called the
quotient vector space
of
V
by
W
.
It is easy to show that the natural projection
V
Æ
V
/
W
that sends a vector in
V
to the coset that it determines is a well-defined surjective linear transformation whose
kernel is
W
.
B.10.4. Theorem.
Let
V
and
W
be vector spaces over a field k. Let
v
1
,
v
2
,...,
v
n
be
a basis for
V
and let
w
1
,
w
2
,...,
w
n
be
any
vectors in
W
. There exists a unique linear
transformation T :
V
Æ
W
such that T(
v
i
) =
w
i
, i = 1, 2,..., n.
Proof.
Easy.
Let k be a field and S a set. Define F(S) to be the set of “formal” sums
Â
r
s
sS
,
Œ
where r
s
Πk and all but a
finite
number of the r
s
are 0. There is an obvious addition
and scalar multiplication making F(S) into a vector space. A more precise definition
of F(S) would define it to be the set of functions j :S Æ k that vanish on all but a finite
number of elements in S. We leave it to the reader to fill in the technical details.
Definition.
The vector space F(S) is called the
free vector space over k with basis S
.
B.10.5. Theorem.
Let S be a set and
V
a vector space. If t : S Æ
V
is any map, then
there is a unique
linear
map T : F(S) Æ V with T(s) = t(s) for all s Œ S.
Proof.
Easy. Like with free groups, the best way to think about this lifting of t to a
map T is via a commutative diagram:
F(S)
T
inclusion map i
»
S
V
t
We have another
universal factorization property
and this is what actually defines F(S)
up to isomorphism.
B.11
Extension Fields
This section describes some fundamental properties of extension fields.
