Math 210
Jerry L. Kazdan
Vectors — and an Application to Least Squares
This brief review of vectors assumes you have seen the basic properties of vectors
previously.
We can write a point in
R
n
as
X
= (
x
1
,...,
x
n
)
. This point is often called a
vector
. Fre
quently it is useful to think of it as an arrow pointing from the origin to the point. Thus,
in the plane
R
2
,
X
= (
1
,

2
)
can be thought of as an arrow from the origin to the point
(
1
,

2
)
.
Algebraic Properties
Alg1.
ADDITION
: If
Y
= (
y
1
,...,
y
n
)
, then
X
+
Y
= (
x
1
+
y
1
,...,
x
n
+
y
n
)
.
Example
: In
R
4
,
(
1
,
2
,

2
,
0
)+(

1
,
2
,
3
,
4
) = (
0
,
4
,
1
,
4
)
.
Alg2.
MULTIPLICATION BY A CONSTANT
:
cX
= (
cx
1
,...,
cx
n
)
.
Example
: In
R
4
, if
X
= (
1
,
2
,

2
,
0
)
, then

3
X
= (

3
,

6
,
6
,
0
)
.
Alg3.
DISTRIBUTIVE PROPERTY
:
c
(
X
+
Y
) =
cX
+
cY
. This is obvious if one writes it
out using components. For instance, in
R
2
:
c
(
X
+
Y
) =
c
(
x
1
+
y
1
,
x
2
+
y
2
) = (
cx
1
+
cy
1
,
cx
2
+
cy
2
) = (
cx
1
,
cx
2
)+(
cy
1
,
cy
2
) =
cX
+
cY
.
Length and Inner Product
IP1.
X
:
=
x
2
1
+
···
+
x
2
n
is the
distance
from
X
to the origin. We will also refer to
X
as the
length
or
norm
of
X
. Similarly
X

Y
is the
distance between X and Y
.
Note that
X
=
0 if and only if
X
=
0, and also that for any constant
c
we have
cX
=

c

X
. Thus,

2
X
=
2
X
=
2
X
.
IP2.
The
inner product
of
X
and
Y
is, by definition,
X
,
Y
:
=
x
1
y
1
+
x
2
y
2
+
···
+
x
n
y
n
.
(1)
(this is also called the
dot product
and written
X
·
Y
). The inner product of two vectors is a
number,
not
another vector. In particular, we have the vital identity
X
2
=
X
,
X
.
Example
: In
R
4
, if
X
= (
1
,
2
,

2
,
0
)
and
Y
= (

1
,
2
,
3
,
4
)
, then
X
,
Y
= (
1
)(

1
) +
(
2
)(
2
)+(

2
)(
3
)+(
0
)(
4
) =

3.
IP3.
ALGEBRAIC PROPERTIES OF THE INNER PRODUCT
. These are obvious from the
above definition of
X
,
Y
.
X
+
Y
,
W
=
X
,
W
+
Y
,
W
,
cX
,
Y
=
c X
,
Y
,
Y
,
X
=
X
,
Y
.
1
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REMARK
: If one works with vectors having complex numbers as elements, then the definition of
the inner product must be modified to
X
,
Y
:
=
x
1
y
1
+
x
2
y
2
+
···
+
x
n
y
n
,
(2)
where
y
1
means the complex conjugate of
y
1
(note: many people put the complex conjugate on the
x
j
instead of the
y
j
). The purpose is to insure that the fundamental property
X
2
=
X
,
X
still
holds. Note, however, that the property
Y
,
X
=
X
,
Y
is now
replaced
by
Y
,
X
=
X
,
Y
.
IP4.
GEOMETRIC INTERPRETATION
:
X
,
Y
=
X
Y
cos
θ
,
where
θ
is the angle between
X
and
Y
. Since cos
(

θ
) =
cos
θ
, the sense in which we measure the angle does not
matter.
To see this, we can restrict our attention to the two dimensional plane containing
X
and
Y
.
Thus, we need consider only vectors in
R
2
. Assume we are not in the trivial case where
X
or
Y
are zero. Let
α
and
β
be the angles that
X
= (
x
1
,
x
2
)
and
Y
= (
y
1
,
y
2
)
make with the
horizontal axis, so
θ
=
β

α
. Then
x
1
=
X
cos
α
and
x
2
=
Y
sin
α
.
Similarly,
y
1
=
Y
cos
β
and
y
2
=
Y
sin
β
. Therefore
X
,
Y
=
x
1
y
1
+
x
2
y
2
=
X
Y
(
cos
α
cos
β
+
sin
α
sin
β
)
=
X
Y
cos
(
β

α
) =
X
Y
cos
θ
.
This is what we wanted.
IP5.
GEOMETRIC CONSEQUENCE
:
X
and
Y
are perpendicular if and only if
X
,
Y
=
0,
since this means the angle
θ
between them is 90 degrees so cos
θ
=
0. We often use the
word
orthogonal
as a synonym for
perpendicular
.
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 Spring '12
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 Linear Algebra, Vectors, Vector Space, Least Squares

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