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1
Example: an orthogonal set
By definition, a set with only one vector is an orthogonal set.
Proof
Let
S
= {
v
1
,
v
2
,
…
,
v
k
}
⊆
R
n
be an orthogonal set and
v
i
≠
0
for
i
= 1, 2,
…
,
k
.
If
c
1
,
c
2
,
…
,
c
k
make
c
1
v
1
+
c
2
v
2
+
"
+
c
k
v
k
=
0
,then
for
i
= 1, 2,
…
,
k
, we have
≠
0
, i.e.,
c
i
= 0.
Definition.
A basis that is an orthogonal set is called an orthogonal
basis.
Example:
The standard basis
E
of
R
n
is an orthogonal basis.
u
u
u
u
u
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Example:
S
= {
v
1
,
v
2
,
v
3
} is an orthogonal basis for
R
3
with
⇒
Proof
⇒
⇒
u
u
u
and
u
u
u
u
u
Proof
By induction on
k
.
The theorem obviously holds for
k
= 1.
Assume the theorem holds for
k
≥
1, and consider the case for
k
+ 1.
We have
1. In the set {
v
1
,
v
2
,
…
,
v
k
,
v
k
+1
},
v
1
,
v
2
,
…
,
v
k
are nonzero orthogonal
vectors, and Span{
v
1
,
v
2
,
…
,
v
k
} = Span{
u
1
,
u
2
,
…
,
u
k
}.
2.
v
k
+1
•
v
i
= 0 for
i
= 1, 2,
…
,
k
, since
3
.
2
1
1
2
1
1
1
2
1
1
2
1
1
1
1
2
1
1
1
1
1
i
k
k
k
k
i
i
i
i
k
i
i
i
i
k
i
i
i
i
k
i
k
i
k
i
k
v
v
v
v
u
v
v
v
v
u
v
v
v
v
u
v
v
v
v
u
v
v
v
v
u
v
u
v
v
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
+
+
+
+
+
+
−
−
−
+
+
+
+
−
−
−
−
−
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.
 Spring '09
 Fong

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