Graphics Reference
In-Depth Information
Definition.
Let
V
be a vector space over a field k. Given a
nonempty
set of vectors
S = {
u
1
,...,
u
n
} in
V
, we define the
span of the set
, span(S), or the
span of the vectors
in S, span(
u
1
,...,
u
n
), to be the set of all vectors that are
linear combinations
of these
vectors, that is,
()
=
(
)
=
{
}
span S
span
uu u
,...,
c
+◊◊◊+
c
u
c
Œ
k
.
1
n
1
1
n
n
i
We say that the set S
spans
X
and the vectors
u
1
,...,
u
n
span
a subspace
X
if
X
=
span S
()
=
span
(
u
1
,...,
u
)
.
n
It is convenient to define span(f) = {
0
} .
It is easy to check that span(
u
1
,...,
u
n
) is a vector subspace of
V
.
Definition.
Let
V
be a vector space over a field k. A
nonempty
set of vectors
S = {
u
1
,...,
u
n
} in
V
, is said to be
linearly dependent
if
c
u
+◊◊◊+
c
nn
u
=
0
11
for some field elements c
1
, c
2
,..., c
k
not all of which are zero. The set S and the
vectors
u
1
,
u
2
,...,
u
n
are said to be
linearly independent
if they are not linearly depen-
dent. It is convenient to define the empty set to be a linearly independent set of vectors.
In other words, vectors are linearly independent if no nontrivial linear combina-
tion of them adds up to the zero vector. Two linearly dependent vectors are often called
collinear
. Two
nonzero
vectors
u
1
and
u
2
are collinear if and only if they are multi-
ples of each other, that is, span(
u
1
) = span(
u
2
).
Definition.
Let
X
be a subspace of a vector space. A set of vectors S is said to be a
basis
for
X
if it is a linearly independent set that spans
X
.
Note.
The definitions of linearly independent/dependent, span, and basis above dealt
only with finite collections of vectors to make the definitions clearer and will apply to
most of our vector spaces. On a few occasions we may have to deal with “infinite dimen-
sional” vector spaces and it is therefore necessary to indicate what changes have to be
made to accommodate those. All one has to do is be a little more careful about what
constitutes a linear combination of vectors. Given an arbitrary, possibly infinite, set of
vectors {
v
a
}
aŒI
in a vector space, define a
linear combination
of those vectors to be a sum
Â
c
v
aa
a
Œ
I
where all but a finite number of the c
a
are zero. With this concept of linear combi-
nation, the definitions above and the next theorem will apply to
all
vector spaces.
B.10.1. Theorem.
Every vector subspace of a vector space has a basis and the
number of vectors in a basis is uniquely determined by the subspace. Every set of lin-
early independent vectors in a vector space can be extended to a basis.
