Linguistic Note 8 Vector Space / July 27, 2007

Linguistic Note

8

Vector Space

　　  TANAKA Akio

1

Additive group V becomes vector space on field K in below condition.

Vector space defines scalar multiplication that is au ∈ V in element a of K and element u of V.

In vector space, arbitrary element of K and V has distributive law, associative law and identity element.

2

Vector space makes functional space in below condition.

Map from set M to field K       K valued function.

All the sets of K valued function on M      F ( M, K )

Definition of addition between two functions f and g      f, g ∈ F ( M, K )     ( f + g ) ( u ) = f ( u ) + g ( u )

Definition of scalar multiplication af between element a on K.     ( af ) ( u ) = af ( u )

Functional space F ( M, K ) becomes vector space.

When set M is M = {1, … , n }, F ( M, K ) is numerical vector space.

3

K–vector space V, W has linear map φ: V → W in below condition.

Arbitrary u, v ∈ V and arbitrary a ∈ K have distributive law and associative law.

Kernel of linear map     Ker (φ) = { u ∈ V | φ ( u ) = 0 }

Image of linear map      Im (φ) = { φ ( u ) |  u ∈ V  }

4

Linear map φ is expressed by matrix.

5

When V and W are finitely dimensional vector spaces, dimension of linear map is below.

Dim ( V ) = dim ( Ker ( φ ) + dim ( Im ( φ ) )

[Note]

Exact sequence on additive group may be helpful for connection of words.

[References]

<On Lineation>

Lineation     Tokyo March 11, 2005

Compendium     Tokyo March 22, 2005

<On connection> 

Language and Spacetime     Generation of Sentence     Tokyo April 13, 2007

Language and Spacetime     Construction of Spacetime     Tokyo April 29, 2007

Tokyo July 27, 2007

Sekinan Research Field of Language