Projections
With the inner product
<x,y>
, we have angles (
<x,y>
=
k
x
k
2
k
y
k
2
cos (
θ
)), and can
speak of orthogonality:
x
⊥
y
⇐⇒
<x,y>
= 0. Here we will consider the standard
inner product for
R
n
:
<x,y>
≡
x
t
y
, but more general inner products can be very
useful in many applications and algorithm development.
If
S
is a subspace of
R
n
(write
S
≤
R
n
), we say
x
⊥
S
if
x
is orthogonal to every
element of
S
. The subspace
S
⊥
≤
R
n
is called the
orthogonal complement
of
S
S
⊥
≡ {
x
∈
R
n
:
x
⊥
S
}
,
and
R
n
=
S
⊕
S
⊥
is a direct sum decomposition of
R
n
into complementary
subspaces in such a way that each
x
∈
R
n
has the unique factorization
x
=
u
+
v
,
with
u
∈
S
and
v
∈
S
⊥
. In this setting,
u
is the
orthogonal projection
of
x
onto
S
.
In this note, we will be looking at a transformation
P
, which satisﬁes
∀
x
∈
R
n
, Px
=
u,
the orthogonal projection of
x
onto
S.
If
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
 Fall '10
 MARK

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