1Vector spaces
IB Linear Algebra
1.3 Direct sums
We are going to define direct sums in many ways in order to confuse students.
Definition
((Internal) direct sum)
.
Suppose
V
is a vector space over
F
and
U, W ⊆ V
are subspaces. We say that
V
is the (internal) direct sum of
U
and
W if
(i) U + W = V
(ii) U ∩ W = 0.
We write V = U ⊕ W .
Equivalently, this requires that every
v ∈ V
can be written uniquely as
u
+
w
with u ∈ U, w ∈ W . We say that U and W are complementary subspaces of V .
You will show in the example sheets that given any subspace
U ⊆ V
,
U
must
have a complementary subspace in V .
Example.
Let
V
=
R
2
, and
U
=
h
0
1
i
. Then
h
1
1
i
and
h
1
0
i
are both
complementary subspaces to U in V .
Definition
((External) direct sum)
.
If
U, W
are vector spaces over
F
, the
(external) direct sum is
U ⊕ W = {(u, w) : u ∈ U, w ∈ W },
with addition and scalar multiplication componentwise:
(u
1
, w
1
) + (u
2
, w
2
) = (u
1
+ u
2
, w
1
+ w
2
), λ(u, w) = (λu, λw).
The difference between these two definitions is that the first is decomposing
V
into smaller spaces, while the second is building a bigger space based on two
spaces.
Note, however, that the external direct sum
U ⊕ W
is the internal direct
sum of
U
and
W
viewed as subspaces of
U ⊕ W
, i.e. as the internal direct sum
of
{
(
u, 0
) :
u ∈ U}
and
{
(
0, v
) :
v ∈ V }
. So these two are indeed compatible
notions, and this is why we give them the same name and notation.
Definition
((Multiple) (internal) direct sum)
.
If
U
1
, ··· , U
n
⊆ V
are subspaces
of V , then V is the (internal) direct sum
V = U
1
⊕ ··· ⊕ U
n
=
n
M
i=1
U
i
if every v ∈ V can be written uniquely as v =
P
u
i
with u
i
∈ U
i
.
This can be extended to an infinite sum with the same definition, just noting
that the sum v =
P
u
i
has to be finite.
For more details, see example sheet 1 Q. 10, where we prove in particular
that dim V =
P
dim U
i
.
Definition
((Multiple) (external) direct sum)
.
If
U
1
, ··· , U
n
are vector spaces
over F, the external direct sum is
U
1
⊕ ··· ⊕ U
n
=
n
M
i=1
U
i
= {(u
1
, ··· , u
n
) : u
i
∈ U
i
},
with pointwise operations.
This can be made into an infinite sum if we require that all but finitely many
of the u
i
have to be zero.