properties of spanning sets

properties of spanning sets

Let

V

be a

vector space

over a field

k

. Let

S

be a subset of

V

. We denote

Sp

⁡

(

S

)

the span of the set

S

. Below are some basic

properties of spanning sets

.

1.

If

S

⊆

T

, then

Sp

⁡

(

S

)

⊆

Sp

⁡

(

T

)

. In particular, if

Sp

⁡

(

S

)

=

V

, every

superset

of

S

spans (

generates

)

V

.

Proof.

If

v

∈

Sp

⁡

(

S

)

, then

v

=

r

1

⁢

v

1

+

⋯

+

r

n

⁢

v

n

for

v

i

∈

S

. But

v

i

∈

T

by

assumption

. So

v

∈

Sp

⁡

(

T

)

as well. If

Sp

⁡

(

S

)

=

V

, and

S

⊆

T

, then

V

=

Sp

⁡

(

S

)

⊆

Sp

⁡

(

T

)

⊆

V

. ∎

2.

If

S

contains

0

, then

Sp

⁡

(

S

-

{

0

}

)

=

Sp

⁡

(

S

)

.

Proof.

Let

T

=

S

-

{

0

}

. So

Sp

⁡

(

T

)

⊆

Sp

⁡

(

S

)

by 1 above. If

v

∈

Sp

⁡

(

S

)

, then

v

=

r

1

⁢

v

1

+

⋯

+

r

n

⁢

v

n

. If one of the

v

i

’s, say

v

i

, is

0

, then

v

=

r

2

⁢

v

2

+

⋯

+

r

n

⁢

v

n

∈

Sp

⁡

(

T

)

. ∎

3.

It is not true that if

S

1

⊇

S

2

⊇

⋯

is a chain of subsets, each

spanning

the same

subspace

W

of

V

, so does their

intersection

.

Proof.

Take

V

=

ℝ

n

, the

Euclidean space

in

n

dimensions

. For each

i

=

1

,

2

,

…

, let

S

i

be the closed ball centered at the origin, with radius

1

/

i

. Then

Sp

⁡

(

S

i

)

=

V

. But the intersection of these

S

i

’s is just the origin, whose span is itself, not

V

.
∎

4.

S

is a basis for

V

iff

S

is a

minimal

spanning set

of

V

. Here, minimal means that any deletion of an element of

S

is no longer a spanning set of

V

.

Proof.

If

S

is a basis for

V

, then

S

spans

V

and

S

is

linearly independent

. Let

T

be the set obtained from

S

with

v

∈

S

deleted. If

T

spans

V

, then

v

can be written as a

linear combination

of

elements

in

T

. But then

S

=

T

∪

{

v

}

would no longer be linearly independent,

contradiction

the assumption. Therefore,

S

is minimal.

Conversely

, suppose

S

is a minimal spanning set for

V

. Furthermore, suppose that

S

is linearly dependent. Let

0

=

r

1

⁢

v

1

+

⋯

⁢

r

n

⁢

v

n

, with

r

1

≠

0

. Then

v

1

=

s

2

⁢

v

2

+

⋯

+

s

n

⁢

v

n

,

(1)

where

s

i

=

-

r

i

/

r

1

. So any linear combination of elements in

S

involving

v

1

can be replaced by a linear combination not involving

v

1

through equation (1). Therefore

Sp

⁡

(

S

)

=

Sp

⁡

(

S

-

{

v

}

)

. But this means that

S

is not minimal, contrary to our assumption. Therefore,

S

must be linearly independent.
∎

Remark

. All of the properties above can be generalized to modules over rings, except the last one, where the

implication

is only one-sided: basis implying minimal spanning set.

Title

properties of spanning sets

Canonical name

PropertiesOfSpanningSets

Date of creation

2013-03-22 18:05:40

Last modified on

2013-03-22 18:05:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Result

Classification

msc 15A03

Classification

msc 16D10

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