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1
LINEAR INDEPENDENCE
DEFINITION
If
S =
{
v
1,
v
2
,…,
v
r
,} is a nonempty set of vectors, then the vector
equation
k
1
v
1
+ k
2
v
2
+ … +
k
r
v
r
=
0
has at least one solution, namely
k
1
=
0,
k
2
=
0,
…
k
r
= 0
If this is the only solution, then
S
is called a
linearly independent
set.
If there are other solutions, then
S
is called a
linearly dependent
set.
EXAMPLE 1
A Linearly Dependent Set
If
v
1
= (2, 1, 0, 3),
v
2
= (1, 2, 5, 1), and
v
3
= (7, 1, 5, 8), then the
set of vectors
S =
{
v
1
,
v
2
,
v
3
} is linearly dependent, since
3
v
1
+
v
2

v
3
= 3(2, 1, 0, 3) + (1, 2, 5, 1) + (7, 1, 5, 8)
(3
ä
2 +1  7, 3
ä
(1) +2 +1, 3
ä
0 + 5  5, 3
ä
3  1  8)
(0, 0, 0, 0) =
0
.
So
3
v
1
+
v
2

v
3
=
0
EXAMPLE 2
A Linearly Dependent Set
The polynomials
p
1
= 1 
x,
p
2
= 5 + 3x  2x
2
,
and
p
3
= 1 + 3
x – x
2
form a linearly dependent set in
P
2
since
3
p
1

p
2
+ 2
p
3
= 3(1 
x
)
–
(5 + 3x  2x
2
) + 2(1 + 3
x – x
2
)
= (3
ä
1  5 + 2) + (3x 3x +6x) + (2x
2
 2x
2
) = 0 + 0 + 0 = 0
So,
3
p
1

p
2
+ 2
p
3
=
0
EXAMPLE 4
Determining Linear Independence/Dependence
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Determine whether the vectors
v
1
, = (1, 2, 3),
v
2
= (5, 6, l),
v
3
= (3, 2, 1)
form a linearly dependent set or a linearly independent set.
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This note was uploaded on 06/20/2008 for the course AM 025b taught by Professor Kiruscheva during the Winter '07 term at UWO.
 Winter '07
 KIRUSCHEVA

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