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are linearly dependent for precisely those values of
λ
for which the equation
xX
1
+
yX
2
+
zX
3
= 0 has a non–trivial solution. This equation is equivalent
to the system of homogeneous equations
λx

y

z
=
0

x
+
λy

z
=
0

x

y
+
λz
=
0
.
Now the coeFcient determinant of this system is
f
f
f
f
f
f
λ

1

1

1
λ

1

1

1
λ
f
f
f
f
f
f
= (
λ
+ 1)
2
(
λ

2)
.
So the values of
λ
which make
X
1
, X
2
, X
3
linearly independent are those
λ
satisfying
λ
6
=

1 and
λ
6
= 2.
5. Let
A
be the following matrix of rationals:
A
=
1
1
2
0
1
2
2
5
0
3
0
0
0
1
3
8
11
19
0
11
.
Then
A
has reduced row–echelon form
B
=
1
0
0
0

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 Fall '10
 Dreibelbis

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