mat67-Homework8_Solutions - MAT067 University of California...

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
.

Solution :
(b) The identical technique can be applied to this part. Applying the Gram-Schmidt
procedure, we form the orthonormal basis { v
1
, v
2
, v
3
} for R 3 as follows:
v
1
=
f
3
f
3
= (0 , 0 , 1) = e
3
,
v
2
=
f
2
− f
2
, u
1
u
1
f
2
− f
2
, u
1
u
1
=
f
2
− u
1
f
2
− u
1
=
(0 , 1 , 0)
√
0
2
+ 1
2
+ 0
2
= (0 , 1 , 0) = e
2
,
v
3
=
f
1
− f
1
, u
1
u
1
− f
1
, u
2
u
2
f
1
− f
1
, u
1
u
1
− f
1
, u
2
u
2
=
f
1
− u
1
− u
2
f
1
− u
1
− u
2
=
(1 , 0 , 0)
√
1 2 + 0 2 + 0 2
= (1 , 0 , 0) = e
1
.
However, this should be clear from the fact that we have started by setting v
1
= e
3
,
then v
2
= e
2
+ e
3
− e
3
, and finally v
3
= e
1
+ e
2
+ e
3
− e
3
− e
2
.
2

2. Let V be a finite-dimensional inner product space over F . Given any vectors u, v ∈ V ,
prove that the following two statements are equivalent:
(a) u, v = 0
(b) u ≤ u + αv for every α ∈ F .
Solution :
Proposition. Let V be a finite-dimensional inner product space over F . Given any
vectors u, v ∈ V , u, v = 0 if and only if u ≤ u + αv for every α ∈ F .
Proof. Let u, v ∈ V be any two vectors in V .
(= ⇒ ) Suppose that u, v = 0, and let α ∈ F . Then
u, αv = α u, v = α 0 = 0
so that u and αv are orthogonal vectors in V . Thus, using that αv 2 ≥ 0 and
applying the Pythagorean Theorem (Theorem 6.3 in the textbook) to u and αv ,
we obtain
u 2 ≤ u 2 + αv 2 = u + αv 2 .