Section 7.3: Systems of Linear Algebraic Equations; Linear Indepen
dence, Eigenvalues, Eigenvectors
Problem 6.
Let
x
(1)
= (1
,
1
,
0)
,
x
(2)
= (0
,
1
,
1)
, and
x
(3)
= (1
,
0
,
1)
. To determine whether these
vectors are linearly independent, we check if the following equation
c
1
x
(1)
+
c
2
x
(2)
+
c
3
x
(3)
=
0
(6.1)
has only the zero solution, i.e.,
(
c
1
, c
2
, c
3
) =
0
. Eq.(6.1) can be expressed as
c
1
1
1
0
+
c
2
0
1
1
+
c
3
1
0
1
=
0
0
0
or, equivalently,
1
0
1
1
1
0
0
1
1
c
1
c
2
c
3
=
0
0
0
.
(6.2)
In terms of the augmented matrix, one has
1
0
1

0
1
1
0

0
0
1
1

0
.
1. Adding
(

1)
times the first row to the second row:
1
0
1

0
0
1

1

0
0
1
1

0
.
2. Adding
(

1)
times the second row to the third row:
1
0
1

0
0
1

1

0
0
0
2

0
.
Hence one has the system
c
1
+
c
3
= 0
,
c
2

c
3
= 0
,
2
c
3
= 0
that is equivalent to Eq.(6.1), the solution of which is
(
c
1
, c
2
, c
3
) =
0
. Hence
x
(1)
= (1
,
1
,
0)
,
x
(2)
= (0
,
1
,
1)
, and
x
(3)
= (1
,
0
,
1)
are linearly independent.
Alternatively, one has
det
1
0
1
1
1
0
0
1
1
= 2
negationslash
= 0
.
1