Chapter 3 Direct Sums, Affine Maps, The Dual Space, Duality

68 Pages·2014·1.15 MB·English

Chapter 3
Direct Sums, A�ne Maps, The Dual
Space, Duality
3.1 Direct Products, Sums, and Direct Sums
There are some useful ways of forming new vector spaces
from older ones.
Definition 3.1. Given p 2 vector spaces E , . . . , E ,
1 p
�
the product F = E E can be made into a vector
1 p
⇥· · ·⇥
space by defining addition and scalar multiplication as
follows:
(u , . . . , u ) + (v , . . . , v ) = (u + v , . . . , u + v )
1 p 1 p 1 1 p p
�(u , . . . , u ) = (�u , . . . ,�u ),
1 p 1 p
for all u , v E and all � .
R
i i i
2 2
With the above addition and multiplication, the vector
space F = E E is called the direct product of
1 p
⇥· · ·⇥
the vector spaces E , . . . , E .
1 p
151
152 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY
The projection maps pr : E E E given by
i 1 p i
⇥· · ·⇥ !
pr (u , . . . , u ) = u
i 1 p i
are clearly linear.
Similarly, the maps in : E E E given by
i i 1 p
! ⇥· · ·⇥
in (u ) = (0, . . . , 0, u , 0, . . . , 0)
i i i
are injective and linear.
It can be shown (using bases) that
dim(E E ) = dim(E ) + + dim(E ).
1 p 1 p
⇥· · ·⇥ · · ·
Let us now consider a vector space E and p subspaces
U , . . . , U of E.
1 p
We have a map
a: U U E
1 p
⇥· · ·⇥ !
given by
a(u , . . . , u ) = u + + u ,
1 p 1 p
· · ·
with u U for i = 1, . . . , p.
i i
2
3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 153
It is clear that this map is linear, and so its image is a
subspace of E denoted by
U + + U
1 p
· · ·
and called the sum of the subspaces U , . . . , U .
1 p
By definition,
U + + U = u + + u u U , 1 i p ,
1 p 1 p i i
· · · { · · · | 2   }
and it is immediately verified that U + + U is the
1 p
· · ·
smallest subspace of E containing U , . . . , U .
1 p
If the map a is injective, then Ker a = 0, which means
that if u U for i = 1, . . . , p and if
i i
2
u + + u = 0
1 p
· · ·
then u = = u = 0.
1 p
· · ·
In this case, every u U + + U has a unique ex-
1 p
2 · · ·
pression as a sum
u = u + + u ,
1 p
· · ·
with u U , for i = 1, . . . , p.
i i
2
154 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY
It is also clear that for any p nonzero vectors u U ,
i i
2
u , . . . , u are linearly independent.
1 p
Definition 3.2. For any vector space E and any p 2
�
subspaces U , . . . , U of E, if the map a defined above is
1 p
injective, then the sum U + + U is called a direct
1 p
· · ·
sum and it is denoted by
U U .
1 p
�· · ·�
The space E is the direct sum of the subspaces U if
i
E = U U .
1 p
�· · ·�
Observe that when the map a is injective, then it is a
linear isomorphism between U U and
1 p
⇥· · ·⇥
U U .
1 p
�· · ·�
The di↵erence is that U U is defined even if
1 p
⇥· · ·⇥
the spaces U are not assumed to be subspaces of some
i
common space.
There are natural injections from each U to E denoted
i
by in : U E.
i i
!
3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 155
Now, if p = 2, it is easy to determine the kernel of the
map a: U U E. We have
1 2
⇥ !
a(u , u ) = u + u = 0 ↵i u = u , u U , u U ,
1 2 1 2 1 2 1 1 2 2
� 2 2
which implies that
Ker a = (u, u) u U U .
1 2
{ � | 2 \ }
Now, U U is a subspace of E and the linear map
1 2
\
u (u, u) is clearly an isomorphism, so Ker a is iso-
7! �
morphic to U U .
1 2
\
As a consequence, we get the following result:
Proposition 3.1. Given any vector space E and any
two subspaces U and U , the sum U + U is a direct
1 2 1 2
sum i↵ U U = 0.
1 2
\
Because of the isomorphism
U U U U ,
1 p 1 p
⇥· · ·⇥ ⇡ �· · ·�
we have
dim(U U ) = dim(U ) + + dim(U ).
1 p 1 p
�· · ·� · · ·
156 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY
If E is a direct sum
E = U U ,
1 p
�· · ·�
since every u E can be written in a unique way as
2
u = u + + u
1 p
· · ·
for some u U for i = 1 . . . , p, we can define the maps
i i
2
⇡ : E U , called projections, by
i i
!
⇡ (u) = ⇡ (u + + u ) = u .
i i 1 p i
· · ·
It is easy to check that these maps are linear and satisfy
the following properties:
⇡ if i = j
i
⇡ ⇡ =
j i
� 0 if i = j,
(
6
⇡ + + ⇡ = id .
1 p E
· · ·
A function f such that f f = f is said to be idempotent.
�
Thus, the projections ⇡ are idempotent.
i
Conversely, the following proposition can be shown:
3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 157
Proposition 3.2. Let E be a vector space. For any
p 2 linear maps f : E E, if
i
� !
f if i = j
i
f f =
j i
� 0 if i = j,
(
6
f + + f = id ,
1 p E
· · ·
then if we let U = f (E), we have a direct sum
i i
E = U U .
1 p
�· · ·�
We also have the following proposition characterizing idem-
potent linear maps whose proof is also left as an exercise.
Proposition 3.3. For every vector space E, if
f : E E is an idempotent linear map, i.e., f f = f,
! �
then we have a direct sum
E = Ker f Im f,
�
so that f is the projection onto its image Im f.
We are now ready to prove a very crucial result relating
the rank and the dimension of the kernel of a linear map.
158 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY
Theorem 3.4. Let f : E F be a linear map. For
!
any choice of a basis (f , . . . , f ) of Im f, let (u , . . . , u )
1 r 1 r
be any vectors in E such that f = f(u ), for i =
i i
1, . . . , r. If s: Im f E is the unique linear map de-
!
fined by s(f ) = u , for i = 1, . . . , r, then s is injective,
i i
f s = id, and we have a direct sum
�
E = Ker f Im s
�
as illustrated by the following diagram:
f
//
Ker f E = Ker f Im s Im f F.
//
� oo s ✓
As a consequence,
dim(E) = dim(Ker f)+dim(Im f) = dim(Ker f)+rk(f).
Remark: The dimension dim(Ker f) of the kernel of a
linear map f is often called the nullity of f.
We now derive some important results using Theorem 3.4.
3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 159
Proposition 3.5. Given a vector space E, if U and
V are any two subspaces of E, then
dim(U) + dim(V ) = dim(U + V ) + dim(U V ),
\
an equation known as Grassmann’s relation.
The Grassmann relation can be very useful to figure out
whether two subspace have a nontrivial intersection in
spaces of dimension > 3.
For example, it is easy to see that in 5, there are sub-
R
spaces U and V with dim(U) = 3 and dim(V ) = 2 such
that U V = 0
\
However, we can show that if dim(U) = 3 and dim(V ) =
3, then dim(U V ) 1.
\ �
As another consequence of Proposition 3.5, if U and V
are two hyperplanes in a vector space of dimension n, so
that dim(U) = n 1 and dim(V ) = n 1, we have
� �
dim(U V ) n 2,
\ � �
and so, if U = V , then
6
dim(U V ) = n 2.
\ �
160 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY
Proposition 3.6. If U , . . . , U are any subspaces of
1 p
a finite dimensional vector space E, then
dim(U + + U ) dim(U ) + + dim(U ),
1 p 1 p
· · ·  · · ·
and
dim(U + + U ) = dim(U ) + + dim(U )
1 p 1 p
· · · · · ·
i↵ the U s form a direct sum U U .
i 1 p
�· · ·�
Another important corollary of Theorem 3.4 is the fol-
lowing result:
Proposition 3.7. Let E and F be two vector spaces
with the same finite dimension dim(E) = dim(F) =
n. For every linear map f : E F, the following
!
properties are equivalent:
(a) f is bijective.
(b) f is surjective.
(c) f is injective.
(d) Ker f = 0.