MAT067
University of California, Davis
Winter 2007
Solutions to Homework Set 8
As usual, we are using
F
to denote either
R
or
C
. We also use
·
,
·
to denote an arbitrary
inner product and
·
to denote its associated norm.
1. Let (
e
1
, e
2
, e
3
) be the canonical basis of
R
3
, and define
f
1
=
e
1
+
e
2
+
e
3
f
2
=
e
2
+
e
3
f
3
=
e
3
.
(a) Apply the Gram-Schmidt process to the basis (
f
1
, f
2
, f
3
).
(b) What do you obtain if you applied the Gram-Schmidt process to the basis (
f
3
, f
2
, f
1
)?
Solution
:
(a) Given the vectors
f
1
= (1
,
1
,
1)
, f
2
= (0
,
1
,
1)
, f
3
= (0
,
0
,
1)
∈
R
3
, we apply
the Gram-Schmidt procedure to form the orthonormal basis
{
u
1
, u
2
, u
3
}
for
R
3
as
follows:
u
1
=
f
1
f
1
=
(1
,
1
,
1)
(1)
2
+ (1)
2
+ (1)
2
=
1
√
3
,
1
√
3
,
1
√
3
,
u
2
=
f
2
−
f
2
, u
1
u
1
f
2
−
f
2
, u
1
u
1
=
f
2
−
2
√
3
u
1
f
2
−
2
√
3
u
1
=
(
−
2
3
,
1
3
,
1
3
)
(
−
2
3
)
2
+
(
1
3
)
2
+
(
1
3
)
2
=
−
2
√
6
,
1
√
6
,
1
√
6
,
u
3
=
f
3
−
f
3
, u
1
u
1
−
f
3
, u
2
u
2
f
3
−
f
3
, u
1
u
1
−
f
3
, u
2
u
2
=
· · ·
=
(
0
,
−
1
2
,
1
2
)
(0)
2
+
(
−
1
2
)
2
+
(
1
2
)
2
=
0
,
−
1
√
2
,
1
√
2
.