A field K can be regarded as a vector space over any subfield F of K
Solution:-
Here K is the set of vectors.
Addition of vectors is the addition composition in the field K. Since K is a field, therefore (K +) is an abelian group.
Further the elements of the subfield F constitute the set of scalars. The composition of scalar multiplication is the multiplication composition in the field K. K is a field,
therefore, a α ∈ K ∀ a ∈ F and ∀ α ∈ K
Because both a and α are element of K. If 1 is the unity element of K, then 1 is also the unity element of the subfield F. We make the following observations.
(i) a(α + β) = aα + aβ, ∀ a ∈ F and α, β ∈ K. This result follows from the left distribution law in K.
(ii) (a + b)α = aα + bα, ∀ a,b ∈ F and ∀ α ∈ K. This result is a consequence of the right distribution law in K.
(iii) (ab) α = a(bα), ∀ a,b ∈ F and ∀ α ∈ K. This result is consequence of associativity of multiplication in K.
(iv) 1α = α ∀ α ∈ K and 1 is the unity element of the subfield F. Since 1 is also the unity element of the field K, therefore 1 α = α ∀ α ∈ K.
Hence, K(F) is a vector space.
Question:-The Vector Space of all ordered n-tuples over a field F
Definition: ⇒Vector Space and Property of Vector Space