Lecture 28
Andrei Antonenko
April 21, 2003
1
Operators
In this lecture we will start studying the most important part of the course on linear algebra
— the theory of operators.
Let
V
be a vector space. Any linear function from
V
to
V
is called the
linear operator
.
We will denote operators by script letters (
A
,
B
,
C
), for example:
A
:
V
→
V.
It means that the operator “rearranges” somehow vectors from the given vector space
V
. An
example of the operator can be an operator of rotation by an angle
α
— applying it, each vector
will be rotated by an angle
α
counterclockwise. Another example — operator of the reflection
with respect to given line — each vector maps to its reflection with respect to some line.
We will learn how to describe operators, and then we will try to classify them, and under
stand actions of them. But first we will study some more facts about vector spaces.
2
Coordinates
Let
V
be a vector space, and assume that
{
e
1
, e
2
, . . . , e
n
}
is a basis. Since it is a basis, we can
represent any other vector
v
as a linear combination of basic vectors, i.e. for any vector
v
we
can find such numbers
a
1
, a
2
, . . . , a
n
, that
v
=
a
1
e
1
+
a
2
e
2
+
· · ·
+
a
n
e
n
.
Such numbers
a
1
, a
2
, . . . , a
n
are called
coordinates
of the vector
v
with respect to basis
{
e
1
, e
2
, . . . , e
n
}
. Let’s consider some examples.
Example 2.1.
Consider the space
R
3
, and let vector
v
= (1
,
1
,
1)
. We will consider 2 different
bases, and find coordinates of
v
with respect to them
1
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•
Let’s consider the standard basis
e
1
= (1
,
0
,
0)
,
e
2
= (0
,
1
,
0)
,
e
3
= (0
,
0
,
1)
.
Then the vector
v
can be represented as
v
= (1
,
1
,
1) = 1(1
,
0
,
0) + 1(0
,
1
,
0) + 1(0
,
0
,
1) = 1
e
1
+ 1
e
2
+ 1
e
3
.
So, the coordinates of this vector with respect to the standard basis are
(1
,
1
,
1)
.
•
Let’s consider another basis
e
0
1
= (0
,
0
,
1)
,
e
0
2
= (0
,
1
,
1)
,
e
0
3
= (1
,
1
,
1)
(we can simply check that this is a basis). Then the vector
v
can be represented as
v
= (1
,
1
,
1) = 0(0
,
0
,
1) + 0(0
,
1
,
1) + 1(1
,
1
,
1) = 0
e
1
+ 0
e
2
+ 1
e
3
.
So, the coordinates of this vector with respect to the basis
{
e
0
1
, e
0
2
, e
0
3
}
are
(0
,
0
,
1)
.
So, we see, that the coordinates of the same vector may be different with the respect to
different bases.
Now we will prove that if the basis is given, the coordinates are defined uniquely.
Theorem 2.2.
Let
V
be a vector space, and let
e
1
, e
2
, . . . , e
n
be a basis. Then for any vector
v
numbers
a
1
, a
2
, . . . , a
n
such that
v
=
a
1
e
1
+
a
2
e
2
+
· · ·
+
a
n
e
n
are defined uniquely, i.e. if
v
=
a
1
e
1
+
a
2
e
2
+
· · ·
+
a
n
e
n
=
b
1
e
1
+
b
2
e
2
+
· · ·
+
b
n
e
n
then
a
1
=
b
1
, a
2
=
b
2
, . . . , a
n
=
b
n
.
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 Spring '03
 Andant
 Linear Algebra, Vector Space, basis, Linear map

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