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]
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Tokyo July 27, 2007
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