Graphics Reference
In-Depth Information
scalar multiplication
, respectively, so that (
V
,+) is an abelian group, and for each a Œ
k and
u
Œ
V
, a·
v
Œ
V
. Furthermore, the following identities hold for each a, b Πk and
u
,
v
Œ
V
:
(1) (distributivity) a·(
u
+
v
) = a·
u
+ a·
v
(2) (distributivity) (a + b)·
u
= a·
u
+ b·
u
(3) (associativity)
(ab)·
u
= a·(b·
u
)
(4) (identity)
1·
u
=
u
For simplicity, the operator · is usually suppressed and one writes a
u
instead of
a·
u
. Also, if the field and the operations are clear from the context, the vector space
(
V
,+,·) will be referred to simply as the vector space
V
. It is easy to show that
-=
()
u
1
u
and it is useful to define a subtraction operators for vectors by setting
-=+
(
.
uvu
v
N-dimensional Euclidean space
R
n
, or
n-space
, is the most well-known example of
a vector space because it is more than just a set of points and admits a well-known
vector space structure over
R
. Namely, let
u
= (u
1
,...,u
n
),
v
= (v
1
,...,v
n
) Œ
R
n
,
c Œ
R
, and define
(
)
uv
u
+=
uv u v
c u u
+
,...,
,...,
+
1
1
n
n
=
(
)
.
1
n
One thinks of elements of
R
n
as either “points” or “vectors,” depending on the context.
More generally, it is easy to see that if k is a field, then k
n
is a vector space over k.
Function spaces are another important class of vector spaces. Let k be a field and
let
A
be a subset of k
n
. Then the set of functions from
A
to k is a vector space over k
by
pointwise addition and scalar multiplication
. More precisely, if f, g :
A
Æ k and c Œ
k, define
f
+
g cf
,
:
A
Æ
k
by
(
)( )
=
()
+
()
( ()
=
(
()
)
.
f
+
g x
f x
g x
and
cf
x
c f x
Definition.
A
subspace
of a vector space
V
is a subset of
V
with the property that it,
together with the induced operations from
V
, is itself a vector space.
For example, under the natural inclusions, the vector spaces
R
m
, m £ n, are all
subspaces of
R
n
.
