Graphics Reference
In-Depth Information
Proof.
See [Dean66].
Theorem B.10.1 justifies the following definition:
Definition.
The
dimension
of a vector space
V
, denoted by dim
V
, is defined to be
the number of vectors in a basis for it. A m-dimensional subspace of an n-dimensional
vector space is said to have
codimension
n-m.
Note.
With our definitions above, the vector space that consists only of the zero
vector has dimension 0 and the empty set is a basis for it. This may sound a little
strange, but it gives us a nice uniform terminology without which some results would
get a little more complicated to state.
Definition.
Let
V
and
W
be vector spaces over a field k. A map T :
V
Æ
W
is called
a
linear transformation
if T satisfies
(
)
=
()
+
()
Ta
vw
+
b
aT
v
bT
w
for all
v
Œ
V
,
w
Œ
W
, and a, b Πk .
B.10.2. Theorem.
The inverse of a linear transformation, if it exists, is a linear
transformation and so is the composite of linear transformations.
Proof.
Easy.
Definition.
A linear transformation is said to be
nonsingular
if it has an inverse;
otherwise, it is said to be
singular
. A nonsingular linear transformation is called a
vector space
isomorphism
.
Definition.
Let
V
and
W
be vector spaces over a field k. If T :
V
Æ
W
is a linear trans-
formation, then define the
kernel
of T, ker (T), and the
image
of T, im (T), by
{
}
()
=Œ
()
=
ker T
vV v
0
T
and
()
=
[
()
]
Õ
im T
T
vvV W
.
Œ
B.10.3. Theorem.
If T :
V
Æ
W
is a linear transformation between vector spaces
V
and
W
, then the kernel and image of T are vector subspaces of
V
and
W
, respectively.
Furthermore,
()
+
()
dim
V
=
dim
im T
dim ker
T
.
Proof.
Easy. See [John67].
Let
W
be a subspace of a vector space
V
over a field k. It is easy to check that the
quotient group
V
/
W
becomes a vector space over k by defining
