�
�
�
�
�
�
1
Lecture
XXX
n-Vectors and
Matrices
n-Vectors
We
define
E
n
to be
the
n
-dimensional
Euclidean
space,
and
R
n
to be the set
of
points
in
E
n
.
Hence
R
n
is
the set of
all
ordered
n
-tuples
of
real
numbers
(
x
1
, . . . , x
n
).
Ordered
n
-tuples
allow repetitions,
i.e.
x
i
=
x
j
is
possible for
i
=
�
j
,
but if
x
i
=
�
x
j
,
then
(
x
1
, . . . , x
i
, . . . , x
j
, . . . , x
n
) = (
x
1
, . . . , x
j
, . . . , x
i
, . . . , x
n
).
These
n
-tuples
can
be
viewed
as
vectors
in
the
n
-dimensional
Euclidean
space.
Hence
we
will
call
them
n-vectors
.
Definition 1
Let
P
= (
p
1
, p
2
, . . . , p
n
)
and
Q
(
q
1
, q
2
, . . . , q
n
)
.
We
say
that
the
distance
between
P
and
Q
is
d
(
P, Q
) =
(
p
1
−
q
1
)
2
+ (
p
2
−
q
2
)
2
+
· · ·
+ (
p
n
−
q
n
)
2
.
We
define
�
0 = (0
,
0
, . . . ,
0).
On
n
-vectors,
we can
define operations
analogous
to the
ones
on
3-vectors:
1.
Multiplication
of
a vector
by a scalar
A
= (
a
1
, a
2
, . . . , a
n
) and
let
k
be a scalar.
Then
k
�
Let
�
A
= (
ka
1
, ka
2
, . . . , ka
n
).
2.
Vector addition
�
�
A
+
�
Let
A
= (
a
1
, a
2
, . . . , a
n
) and
B
= (
b
1
, b
2
, . . . , b
n
).
Then
�
B
= (
a
1
+
b
1
, a
2
+
b
2
, . . . , a
n
+
b
n
).
3.
Dot product
�
�
A
·
�
Let
A
= (
a
1
, a
2
, . . . , a
n
) and
B
= (
b
1
, b
2
, . . . , b
n
).
Then
�
B
=
a
1
b
1
+
a
2
b
2
+
+
a
n
b
n
.
· · ·
4.
Magnitude
of