Natural Language Processing

78 Natural Language Processing: Semantic Aspects

Proposition 4.4.4. Let U

1

and U

2

be linear subspaces of V. Then

V = U

1

 U

2

if and only if the following holds true:

(1) V = U

1

+ U

2

(2) U

1

 U

2

= {0}.

Proof. (²) Suppose V = U

1

 U

2

, then V = U

1

+ U

2

. Let v ¢ U

1

U

2

.

Then

N

NN

N

0 0 .

UU

UU

ŒŒ

ŒŒ

=+=+vv v

21

12

Hence v has two different representations as a sum of vectors

from U

1

and U

2

. It follows v = 0 and U

1

U

2

= {0}.

(°) Let us suppose that (1) and (2) are true. Then, for every v ¢ V,

this vector has a decomposition as

v = u

1

+ u

2

, u

1

¢ U

1

, u

2

¢ U

2

.

For the uniqueness, suppose

v = u'

1

+ u'

2

, with u'

1

¢ U

1

, u'

2

¢ U

2

.

Substracting these equalities, we have 0 = (u

1

– u'

1

) + (u

2

– u'

2

).

It follows u

1

– u'

1

= u'

2

– u

2

¢ U

1

 U

2

. Since U

1

 U

2

= {0}, we have ...