Ex 4 Is
( )
23 2 3
4, , :Wxxxandxarerealnumbers=
a subspace of
3
VR=
?
Theorem 4.6 The Intersection of Two Subspaces Is a Subspace
If
V
and
W
are both subspaces of a vector space
,U
then the intersection of
V
and
W
(denoted by
VW
) is also a
subspace of
.U
4.4 Spanning Sets and Linear Independence
Definition of a Linear Combination of Vectors
A vector
v
in a vector space
V
is a linear combination of the vectors
12
, ,....., k
uu u
in
V
when
v
can be written in the
form
11 2 2 ..... kk
vcu c cu=+ ++u
where
12
, ,....., k
cc c
are scalars.
Ex 1 Write the vector
( )
49 99 19
24 2
,,14,−
as a linear combination of the vectors in
( ) ( )
6, 7, 8, 6 , 4, 6, 4, 1S=− −
if
possible.
Definition of a Spanning Set of a Vector Space
Let
12
,,.....,
k
Svv v=
be a subset of a vector space
.V
The set
S
is a spanning set of
V
when every vector in
V
can
be written as a linear combination of vectors in
.S
In such cases it is said that
S
spans
.V
i.e. If every vector in a vector space
V
can be written as a linear combination of vectors in a set
S
, then
S
is a spanning
set of the vector space
.V
