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MATH1850U/2050U:
Chapter 6 cont…
1
INNER PRODUCT SPACES cont…
Angle and Orthogonality in Inner Product Spaces (6.2; pg. 345)
Recall:
In sections 3.23.3, you already learned about the CauchySchwartz inequality,
properties of lengths and distances, cosine of an angle between 2 vectors, and the
Pythagorean Theorem for
R
n
.
Definition:
Two vectors
u
and
v
in an inner product space are called
orthogonal
if
0
,
v
u
.
Example:
Suppose
p
,
q
, and
r
are defined as given below.
i)
Are
p
and
q
orthogonal?
ii)
Are
p
and
r
orthogonal?
Definition:
Let
W
be a subspace of an inner product space
V
.
A vector
u
in
V
is said to
be
orthogonal
to
W
if it is orthogonal to every vector in
W
.
The set of all vectors in
V
that are orthogonal to
W
is called the
orthogonal complement
of
W
.
This set is denoted
W
.
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View Full DocumentMATH1850U/2050U:
Chapter 6 cont…
2
Theorem (Properties of Orthogonal Complements):
If
W
is a subspace of an inner
product space
V
, then:
a)
W
is a subspace of
V
.
b)
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 Spring '11
 PaulaTu
 Math

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