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.