free vector space over a set

free vector space over a set

In this entry we

construct

the

free vector space over a set

, or
the

vector space

generated by a set

[

1

]

.
For a set

X

, we shall denote this vector space by

C

⁢

(

X

)

.
One application of this construction is given in

[

2

]

,
where the free vector space is used to define the

tensor product

for
modules.

To define the vector space

C

⁢

(

X

)

, let us first define

C

⁢

(

X

)

as a
set. For a set

X

and a field

𝕂

, we define

C

⁢

(

X

)

=

{

f

:

X

→

𝕂

|

f

-

1

(

𝕂

\

{

0

}

)

is finite

}

.

In other words,

C

⁢

(

X

)

consists of functions

f

:

X

→

𝕂

that are non-zero only
at finitely many points in

X

.
Here, we denote the

identity element

in

𝕂

by

1

, and
the

zero element

by

0

.
The vector space structure for

C

⁢

(

X

)

is defined as follows. If

f

and

g

are
functions in

C

⁢

(

X

)

, then

f

+

g

is the mapping

x

↦

f

⁢

(

x

)

+

g

⁢

(

x

)

.
Similarly, if

f

∈

C

⁢

(

X

)

and

α

∈

𝕂

, then

α

⁢

f

is the mapping

x

↦

α

⁢

f

⁢

(

x

)

. It is not difficult to
see that these operations are well defined, i.e., both

f

+

g

and

α

⁢

f

are again functions in

C

⁢

(

X

)

.

0.0.1

Basis for

C

⁢

(

X

)

If

a

∈

X

,
let us define the function

Δ

a

∈

C

⁢

(

X

)

by

Δ

a

⁢

(

x

)

=

{

1

when

⁢

x

=

a

,

0

otherwise.

These functions form a

linearly independent

basis for

C

⁢

(

X

)

, i.e.,

C

⁢

(

X

)

=

span

{

Δ

a

}

a

∈

X

.

(1)

Here, the space

span

{

Δ

a

}

a

∈

X

consists of all
finite

linear combinations

of elements in

{

Δ

a

}

a

∈

X

.
It is clear that any element in

span

{

Δ

a

}

a

∈

X

is a member in

C

⁢

(

X

)

.
Let us check the other

direction

. Suppose

f

is a member in

C

⁢

(

X

)

.
Then, let

ξ

1

,

…

,

Ã

⁢



⁢

Â

⁢

⁢

ξ

N

be the distinct points in

X

where

f

is non-zero.
We then have

f

=

∑

i

=

1

N

f

⁢

(

ξ

i

)

⁢

Δ

ξ

i

,

and we have established equality in

equation

1

.

To see that the set

{

Δ

a

}

a

∈

X

is linearly independent,
we need to show that its any finite subset is linearly independent.
Let

{

Δ

ξ

1

,

…

,

Δ

ξ

N

}

be such
a finite subset, and
suppose

∑

i

=

1

N

α

i

⁢

Δ

ξ

i

=

0

for some

α

i

∈

𝕂

. Since the points

ξ

i

are pairwise distinct, it
follows that

α

i

=

0

for all

i

. This shows that the set

{

Δ

a

}

a

∈

X

is linearly independent.

Let us define the mapping

ι

:

X

→

C

⁢

(

X

)

,

x

↦

Δ

x

.
This mapping gives a bijection between

X

and the basis
vectors

{

Δ

a

}

a

∈

X

. We can thus identify these
spaces. Then

X

becomes a linearly independent basis for

C

⁢

(

X

)

.

0.0.2

Universal property of

ι

:

X

→

C

⁢

(

X

)

The mapping

ι

:

X

→

C

⁢

(

X

)

is

universal

in
the following sense. If

ϕ

is an arbitrary mapping from

X

to a
vector space

V

, then there exists a unique mapping

ϕ

¯

such that the below

diagram

commutes:

\xymatrix

⁢

X

⁢

\ar

⁢

[

r

]

ϕ

⁢

\ar

⁢

[

d

]

ι

⁢

&

⁢

V

⁢

C

⁢

(

X

)

⁢

\ar

⁢

[

u

⁢

r

]

ϕ

¯

⁢

&

Proof.

We define

ϕ

¯

as the

linear mapping

that
maps the basis elements of

C

⁢

(

X

)

as

ϕ

¯

⁢

(

Δ

x

)

=

ϕ

⁢

(

x

)

.
Then, by definition,

ϕ

¯

is linear. For uniqueness,
suppose that there are linear mappings

ϕ

¯

,

σ

¯

:

C

⁢

(

X

)

→

V

such that

ϕ

=

ϕ

¯

∘

ι

=

σ

¯

∘

ι

.
For all

x

∈

X

, we then have

ϕ

¯

⁢

(

Δ

x

)

=

σ

¯

⁢

(

Δ

x

)

.
Thus

ϕ

¯

=

σ

¯

since both mappings are linear and
the coincide on the basis elements.

□

References

1

W. Greub,

Linear Algebra

,
Springer-Verlag, Fourth edition, 1975.

2

I. Madsen, J. Tornehave,

From Calculus to

Cohomology

,
Cambridge University press,
1997.

Title

free vector space over a set

Canonical name

FreeVectorSpaceOverASet

Date of creation

2013-03-22 13:34:34

Last modified on

2013-03-22 13:34:34

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15-00

Synonym

vector space generated by a set

Related topic

TensorProductBasis

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