Let (F, +, ∙) be field. The elements of F will be called scalars. Let V be a non-empty set whose elements will be called vectors. Then V is a vector space over the field F, if
There is defined an internal composition in V called addition of vector field and denoted by ‘+’. Also for this composition V is an abelian group.
There is an external composition in V F called scalar multiplication and denoted multiplicatively i.e., aα `in` V for all α`in` F and for all α `in` V. in other words V is closed with respect to scalar multiplication.
The two composition i.e., scalar multiplication and addition of vectors satisfy the following postulates
(i) a (α + β) = aα + aβ `AA`a `in` F and `AA` α, β `in` V
(ii) (a + b) α = aα + bα `AA` a, b `in` F and `AA` α `in` V
(iii) (ab) α = a (bα) `AA` a, b `in` F and `AA` α `in` V
(iv) 1α = α `AA` α `in` V and 1 is the vanity element of the field F.
When V is vector space over the field F, we shall say that V(F) is a vector space. If the field F is understood we van simply say that V is a vector space.
Example for Vector Field Theory
Show that a field F may be considered as a vector space over F if scalar multiplication is identified with field multiplication
Solution
Let (F, +, ∙) is a field. Take F as the set of vectors and also as the set of scalars. Take the addition operation on the field F as the addition of vectors and the multiplication operation on the field F as the operation of scalar multiplication i.e., multiplication of a vector by a scalar. Then F is a vector space over F as shown below I have recently faced lot of problem while learning how big is 8mm, But thank to online resources of math which helped me to learn myself easily on net.
Since (F, +, ∙) is a field, therefore (f, +) is an abelian group
Further if a, b are any scalars i.e., a, b `in` F and α, β are any vectors i.e., α, β `in` F then
a (α + β) = aα + aβ and (a + b) α = aα + bα
These results follow from the distributive laws in the field F.
Also (ab) α = a (bα) because the multiplication on the field F is associative
Also if 1 is the unity element of the field F and α is any vector i.e., α `in` F, then 1α = α
Hence F(F) is a vector space
There is defined an internal composition in V called addition of vector field and denoted by ‘+’. Also for this composition V is an abelian group.
There is an external composition in V F called scalar multiplication and denoted multiplicatively i.e., aα `in` V for all α`in` F and for all α `in` V. in other words V is closed with respect to scalar multiplication.
The two composition i.e., scalar multiplication and addition of vectors satisfy the following postulates
(i) a (α + β) = aα + aβ `AA`a `in` F and `AA` α, β `in` V
(ii) (a + b) α = aα + bα `AA` a, b `in` F and `AA` α `in` V
(iii) (ab) α = a (bα) `AA` a, b `in` F and `AA` α `in` V
(iv) 1α = α `AA` α `in` V and 1 is the vanity element of the field F.
When V is vector space over the field F, we shall say that V(F) is a vector space. If the field F is understood we van simply say that V is a vector space.
Example for Vector Field Theory
Show that a field F may be considered as a vector space over F if scalar multiplication is identified with field multiplication
Solution
Let (F, +, ∙) is a field. Take F as the set of vectors and also as the set of scalars. Take the addition operation on the field F as the addition of vectors and the multiplication operation on the field F as the operation of scalar multiplication i.e., multiplication of a vector by a scalar. Then F is a vector space over F as shown below I have recently faced lot of problem while learning how big is 8mm, But thank to online resources of math which helped me to learn myself easily on net.
Since (F, +, ∙) is a field, therefore (f, +) is an abelian group
Further if a, b are any scalars i.e., a, b `in` F and α, β are any vectors i.e., α, β `in` F then
a (α + β) = aα + aβ and (a + b) α = aα + bα
These results follow from the distributive laws in the field F.
Also (ab) α = a (bα) because the multiplication on the field F is associative
Also if 1 is the unity element of the field F and α is any vector i.e., α `in` F, then 1α = α
Hence F(F) is a vector space
