dimension formulae for vector spaces

dimension formulae for vector spaces

In this entry we look at various formulae
involving the

dimension

of a

vector space

.

Throughout this entry,

K

will be a field,
and

V

and

W

will be vector spaces over

K

.
The dimension of a vector space

U

over

K

will be denoted by

dim

⁡

(

U

)

,
or by

dim

K

⁡

(

U

)

if the

ground field

needs to be emphasized.

All of these formulae potentially involve

infinite

cardinals

,
so the reader should have a basic knowledge of

cardinal arithmetic

in order to understand them in full generality.

Subspaces

If

S

and

T

are

subspaces

of

V

, then

dim

⁡

(

S

)

+

dim

⁡

(

T

)

=

dim

⁡

(

S

∩

T

)

+

dim

⁡

(

S

+

T

)

.

Rank-nullity theorem

The

rank-nullity theorem

states that if

ϕ

:

V

→

W

is a

linear mapping

,
then the dimension of

V

is the sum of the dimensions of the image and kernel of

ϕ

:

dim

⁡

(

V

)

=

dim

⁡

(

Im

⁡

ϕ

)

+

dim

⁡

(

Ker

⁡

ϕ

)

.

In particular, if

U

is a subspace of

V

then

dim

⁡

(

V

)

=

dim

⁡

(

V

/

U

)

+

dim

⁡

(

U

)

.

The rank-nullity theorem can also be stated in terms of

short exact sequences

:
if

0

⟶

U

⟶

V

⟶

W

⟶

0

is a short exact sequence of vector spaces over

K

,
then

dim

⁡

(

V

)

=

dim

⁡

(

U

)

+

dim

⁡

(

W

)

.

This can be generalized to infinite

exact sequences

:
if

⋯

⟶

V

n

+

1

⟶

V

n

⟶

V

n

-

1

⟶

⋯

is an exact sequence of vector spaces over

K

,
then

∑

n

⁢

even

dim

⁡

(

V

n

)

=

∑

n

⁢

odd

dim

⁡

(

V

n

)

.

(This is indeed a

generalization

,
because any finite exact sequence of vectors spaces
terminating with

0

at both ends
can be considered as an infinite exact sequence
in which all remaining terms are

0

.)

Direct sums

If

(

V

i

)

i

∈

I

is a family of vector spaces over

K

,
then

dim

⁡

(

⊕

i

∈

I

V

i

)

=

∑

i

∈

I

dim

⁡

(

V

i

)

.

Cardinality of a vector space

The

cardinality

of a vector space
is determined by its dimension and the cardinality of the ground field:

|

V

|

=

{

|

K

|

dim

⁡

(

V

)

,

if

⁢

dim

⁡

(

V

)

⁢

is finite

;

max

⁡

{

|

K

|

,

dim

⁡

(

V

)

}

,

if

⁢

dim

⁡

(

V

)

⁢

is infinite

.

The effect of the above

formula

is somewhat different
depending on whether

K

is finite (

http://planetmath.org/FiniteField

) or infinite.
If

K

is finite, then it reduces to

|

V

|

=

{

|

K

|

dim

⁡

(

V

)

,

if

⁢

dim

⁡

(

V

)

⁢

is finite

;

dim

⁡

(

V

)

,

if

⁢

dim

⁡

(

V

)

⁢

is infinite

.

If

K

is infinite, then it can be expressed as

|

V

|

=

{

1

,

if

⁢

dim

⁡

(

V

)

=

0

;

|

K

|

,

if

⁢

0

<

dim

⁡

(

V

)

≤

|

K

|

;

dim

⁡

(

V

)

,

if

⁢

dim

⁡

(

V

)

≥

|

K

|

.

Change of ground field

If

F

is a subfield of

K

,
then

V

can be considered as a vector space over

F

.
The dimensions of

V

over

K

and

F

are related by the formula

dim

F

(

V

)

=

[

K

:

F

]

⋅

dim

K

(

V

)

.

In this formula,

[

K

:

F

]

is the degree of the field extension

K

/

F

,
that is, the dimension of

K

considered as a vector space over

F

.

Space of functions into a vector space

If

S

is any set,
then the set

K

S

of all

functions

from

S

into

K

becomes a vector space over

K

if we define the

operations

pointwise

,
that is,

(

f

+

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

and

(

λ

⁢

f

)

⁢

(

x

)

=

λ

⁢

f

⁢

(

x

)

for all

f

,

g

∈

K

S

, all

x

∈

S

, and all

λ

∈

K

.
The dimension of this vector space is given by

dim

⁡

(

K

S

)

=

{

|

S

|

,

if

⁢

S

⁢

is finite

;

|

K

|

|

S

|

,

if

⁢

S

⁢

is infinite

.

The case where

S

is infinite is not straightforward to prove.
Proofs can be found in books by Baer

[

1

]

and Jacobson

[

2

]

,
among others.

More generally, we can consider the space

V

S

,
which is really just the

direct product

(

http://planetmath.org/DirectProduct

) of copies of

V

indexed by

S

.
We get

dim

⁡

(

V

S

)

=

{

0

,

if

⁢

dim

⁡

(

V

)

=

0

;

|

S

|

⋅

dim

⁡

(

V

)

,

if

⁢

S

⁢

is finite

;

|

V

|

|

S

|

,

otherwise

.

Dual space

Given any basis

B

of

V

,
the

dual space

V

*

is

isomorphic

to

K

B

via the

mapping

f

↦

f

|

B

.
So the formula of the previous

section

can be applied to give a formula for the dimension of

V

*

:

dim

⁡

(

V

*

)

=

{

dim

⁡

(

V

)

,

if

⁢

dim

⁡

(

V

)

⁢

is finite

;

|

K

|

dim

⁡

(

V

)

,

if

⁢

dim

⁡

(

V

)

⁢

is infinite

.

In particular, this formula implies that

V

is isomorphic to

V

*

if and only if

V

is finite-dimensional.
(Students who are familiar with the fact that
an infinite-dimensional

Banach space

can be isomorphic to its dual
are sometimes surprised to learn that an infinite-dimensional vector space
cannot be isomorphic to its dual,
for a Banach space is surely a vector space.
But the term

dual

is used in different senses in these two statements,
so there is no

contradiction

.
In the theory of Banach spaces
one is usually only interested in the

continuous

linear functionals

,
and the resulting ‘continuous’ dual
is a subspace of the full dual used in the above formula.)

Space of linear mappings

The set

Hom

K

⁡

(

V

,

W

)

of all linear mappings from

V

into

W

is itself a vector space over

K

,
with the operations defined in the obvious way,
namely

(

f

+

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

and

(

λ

⁢

f

)

⁢

(

x

)

=

λ

⁢

f

⁢

(

x

)

for all

f

,

g

∈

Hom

K

⁡

(

V

,

W

)

, all

x

∈

V

, and all

λ

∈

K

.
The dual space

V

*

=

Hom

K

⁡

(

V

,

K

)

considered in the previous section
is a special case of this.
For any basis

B

of

V

, the mapping

f

↦

f

|

B

defines an

isomorphism

between

Hom

K

⁡

(

V

,

W

)

and

W

B

,
so that from an earlier section we get

dim

⁡

(

Hom

K

⁡

(

V

,

W

)

)

=

{

0

,

if

⁢

dim

⁡

(

W

)

=

0

;

dim

⁡

(

V

)

⋅

dim

⁡

(

W

)

,

if

⁢

dim

⁡

(

V

)

⁢

is finite

;

|

W

|

dim

⁡

(

V

)

,

otherwise

.

In the special case

W

=

V

this can be simplified to

dim

⁡

(

End

K

⁡

(

V

)

)

=

{

dim

⁡

(

V

)

2

,

if

⁢

dim

⁡

(

V

)

⁢

is finite

;

|

K

|

dim

⁡

(

V

)

,

otherwise

.

Tensor products

The dimension of the

tensor product

(

http://planetmath.org/TensorProduct

) of

V

and

W

is given by

dim

⁡

(

V

⊗

W

)

=

dim

⁡

(

V

)

⋅

dim

⁡

(

W

)

.

Banach spaces

The dimension of a Banach space,
considered as a vector space,
is sometimes called the

Hamel dimension

,
in order to distinguish it from other

concepts

of dimension.
For an infinite-dimensional Banach space

B

we have

dim

⁡

(

B

)

=

|

B

|

.

The tricky part of establishing this formula
is to show that the dimension is always
at least the cardinality of the

continuum

.
A short proof of this is given in a paper by Lacey

[

3

]

.

The above formula suggests that Hamel dimension
is not a very useful concept for infinite-dimensional Banach spaces,
which is indeed the case.
Nonetheless, it is interesting to see how Hamel dimension relates to
the usual concept of dimension in

Hilbert spaces

.
If

H

is a Hilbert space,
and

d

is its dimension
(meaning the cardinality of an

orthonormal basis

),
then the Hamel dimension

dim

⁡

(

H

)

is given by

dim

⁡

(

H

)

=

{

d

,

if

⁢

d

⁢

is finite

;

d

ℵ

0

,

if

⁢

d

⁢

is infinite

.

References

1

Reinhold Baer,

Linear Algebra

and Projective Geometry

,
Academic Press, 1952.

2

Nathan Jacobson,

Lectures in

Abstract Algebra

,
Volume II:

Linear Algebra

,
D. Van Nostrand Company Inc., 1953.

3

H. Elton Lacey,

The Hamel Dimension of any Infinite Dimensional

Separable

Banach Space is c

,
Amer. Math. Mon. 80 (1973), 298.

Title

dimension formulae for vector spaces

Canonical name

DimensionFormulaeForVectorSpaces

Date of creation

2013-03-22 16:31:19

Last modified on

2013-03-22 16:31:19

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

25

Author

yark (2760)

Entry type

Feature

Classification

msc 15A03

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