Example 2: A linearly dependent set
Find out if the set { ( 1, 0, 3 ), ( -2, 6, -9 ), ( 1, -2, 4 ) } = {
1
,
2
,
3
} of vectors of
R
3
is linearly independent.
Solution:
a
1
1
+ a
2
2
+ a
3
3
=
<=>
a
1
1
0
3
+
a
2
-2
6
-9
+
a
3
1
-2
4
=
0
0
0
<=>
1
-2
1
0
6
-2
3
-9
4
a
1
a
2
a
3
=
0
0
0
We use
Gaussian elimination
:
=>
1
-2
1
0
0
6
-2
0
3
-9
4
0
~
1
-2
1
0
0
3
-1
0
0
-3
1
0
~
1
-2
1
0
0
3
-1
0
0
0
0
0
-Add the 1st row multiplied by -3 to the 3rd row
-Multiply the 2nd row by ½
-Add the 2nd row to the 3rd row
-Solve the upper triangular system by backward substitution:
a
1
-2
a
2
+
a
3
= 0
3
a
2
-
a
3
= 0
a
3
=
t
C
<=> (for example)
a
3
= 3
a
2
= 1
a
1
= -1
=> -1 ( 1, 0, 3 ) +1 ( -2, 6, -9 ) +3 ( 1, -2, 4 ) =
=> the set is linearly dependent
Go back to theory
1 + a2 
C