MA 265 LECTURE NOTES: FRIDAY, MARCH 28
Inner Product Spaces
Length and Direction in Inner Product Spaces.
Let
(
V,
+
,
·
,
(
,
)
)
be an inner product space. We
deﬁne length, distance, and direction in the following way:
•
The
length
of a vector
v
∈
V
is given by

v

=
p
(
v
,
v
).
•
The
distance
between two vectors
u
,
v
∈
V
is given by
d
(
u
,
v
) =

v

u

=
p
(
v

u
,
v

u
) =
p

u

2

2 (
u
,
v
) +

v

2
.
•
The
angle
between two nonzero vectors
u
,
v
∈
V
is given by
θ
= arccos
(
u
,
v
)

u

v

where
0
≤
θ
≤
π.
By (Positivity), we see that

v

= 0 if and only if
v
=
0
. By (Linearity), we see that

c
v

=

c

v

for
any scalar
c
∈
R
. Using these deﬁnitions, we say that two vectors
u
and
v
– not necessarily nonzero! – are
•
in the
same direction
if (
u
,
v
) =

u
 
v

;
•
orthogonal
or
perpendicular
if (
u
,
v
) = 0; and
•
in
opposite directions
if (
u
,
v
) =

u
 
v

.
Orthonormal Sets.
Let
(
V,
+
,
·
,
(
,
)
)
be an inner product space. Say that we have a collection
S
=
{
v
1
,
v
2
, ...,
v
n
}
of vectors from
V
. We say that
S
is
orthogonal
if we have (
v
i
,
v
j
) = 0 for all
i
6
=
j
. We
say that a vector
v
is
normal
if each it has length 1 i.e.,

v

= 1. We use the portmanteau
orthonormal
if
S
is orthogonal and each vector in
S
is normal.
Let