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Lecture 33
Andrei Antonenko
May 2, 2003
1 Direct sum of vector spaces
In the ﬁrst part of this lecture we will consider some more concepts from the theory of vector
spaces.
Deﬁnition 1.1.
Sum
of two vector spaces
U
and
V U
+
V
is a vector space which consists of
all vectors
u
+
v
, where
u
∈
U
and
v
∈
V
.
Deﬁnition 1.2.
Vector spaces
U
1
,U
2
,...,U
n
are called
linearly independent
if from
u
1
+
···
+
u
n
= 0
, where
u
i
∈
U
i
it follows that
u
i
= 0
for all
i
.
Sum of the linearly independent vector spaces is called a
direct sum
of these vector spaces
and is denoted by
U
1
⊕ ··· ⊕
U
n
.
Deﬁnition 1.3.
The vector space
V
is said to be equal to a
direct sum
of vector spaces
U
1
,...,U
n
V
=
U
1
⊕ ··· ⊕
U
n
if any vector
v
from
V
can be represented as
v
=
u
1
+
···
+
u
n
,
u
i
∈
U
i
uniquely.
For example, the plane
R
2
is equal to a direct sum of
x

and
y

axes.
Example 1.4.
The space of all matrices is equal to a direct sum of the space of all symmetric
matrices and all skewsymmetric matrices, since any matrix
A
can be uniquely represented as
A
=
A
+
A
>
2
+
A

A
>
2
,
and one can check that
A
+
A
>
2
is always symmetric, and
A

A
>
2
is always skewsymmetric. More
over, the sum is direct, since if a matrix is both symmetric and skewsymmetric, it is equal to
0
matrix.
1
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View Full Document2 Invariant spaces
Deﬁnition 2.1.
Let
A
be an operator in vector space
V
. The subspace
U
⊂
V
is called an
invariant subspace
with respect to operator
A
if
A
U
⊂
U.
This deﬁnition means, that the vectors from invariant subspace remain in this subspace
after application of the operator
A
.
Example 2.2.
Considering the operator of rotation in the 3dimensional space around some
axes, we can see, that all the planes, perpendicular to the axes of rotation are invariant. More
over, the axes of rotation is invariant itself.
If the basis
{
e
1
,...e
n
}
of
V
is such that ﬁrst
k
vectors
{
e
1
,...,e
k
}
is a basis of
U
, then the
matrix of the operator
A
in this basis has the following form:
ˆ
B D
0
C
!
.
Moreover, if the space
V
is equal to a direct sum of two subspaces
V
=
U
⊕
W
, and
{
e
1
,...,e
k
}
is a basis of
U
, and
{
e
k
+1
,...,e
n
}
is a basis of
W
, then the matrix of
A
has the following form:
ˆ
B
0
0
C
!
.
Example 2.3.
Consider the rotation in the 3dimensional space about some axes be an angle
α
. In the basis
{
e
1
,e
2
,e
3
}
, if the vector
e
3
is directed along the axes of rotation, the matrix of
this operator has the following form:
cos
α

sin
α
0
sin
α
cos
α
0
0
0
1
This matrix is consistent with the decomposition of
R
3
into a direct sum of two invariant
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 Spring '03
 Andant

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