ERG 2018 Tutorial 4
September 30, 2008
1. Vector space, subspace, Span and linear independence
For the strict definition of
vector space
and
subspace
, you can refer to textbook P.189 and P.197.
Let
12
,,,
k
v
vv
K
be vectors in a vector space V.
A vector
v
in V is called a
linear combination
of
k
v
K
if
1
1
22
1
k
k
k
jj
j
v
a
v
a
v
a
v
av
=
=+
+
+=
∑
L
Consider
{ }
k
S
v
=
K
as a set of vectors in V, then
span S
is the set of all vectors in V that can
be written as linear combinations of
k
v
K
.
The vectors
k
v
K
are said to be
linearly independent
if, whenever
1
1
0
kk
a
v
a
v
+
+
L
,
then
0
k
a
aa
====
L
. Otherwise, they are
linearly dependent
.
Note that
span
k
v
K
is a subspace of V (P.210)
1.1 Example: Linearly Independent (LI)
Determine whether or not the set
123
{,,}
vvv
is linearly dependent, where
1
21
2
,
1
,1
312

==
=

2. The Basic Unit Vectors in 3D Vector Space (V=
3
¡
)
100
0
,
1
,0
001
ijk
===
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 Spring '10
 Chan
 Linear Algebra, Vector Space, vk

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