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Vector Space

Let

then,

then

The elements of

**< F, +, . >**be a field, the elements of**F**are known as scalars. Let**V**be a non-empty set whose elements are denoted by**a, b, c**, etc. and let there be two compositions on**V**known as internal composition {denoted by**+**) and scalar multiplication (external composition denoted by**‘∴’**) Then the set**V**together with these two compositions is said to form a vector space over the field**F**, to be denoted by**V(F)**, if the following axioms are satisfied:**V**_{1}: Closure property: ∀ a, b V a + b V.**V**_{2}: Associative law: ∀ a, b, c V (a + b) + c = a (b + c).**There exists an element**

VV

_{3}: Existence of identity:**0 V**such that**a + 0 = a = 0 + a ∀ a V**then,

**0**is called the identity element.**V**, there exists_{4}: Existence of inverse: ∀ a V**b V**such that**a + b = b + a = 0**.then

**b**is known as the inverse of**a**and vice versa.**V**Scalar multiplication is associative, i.e._{5}:**(****a) = (****)a ∀****,****F and****V**.**V**_{6}: Commutative law: a + b = b + a**V**Scalar multiplication is distributive over addition in_{7}:**F**.**(a + b) = a + b, ∀ a, b V and ∀ F**.**V**Distributivity of scalar multiplication over addition in_{8}:**F**.**(****+)a =****a + a ∀****, F and ∀ a V****V**Property of unity Let_{9}:**1 F**be the unity of**F**, then**1.a = a = a.1; ∀ a V**.The elements of

**F**are known as scalars and the elements of**V**are called vectors.**Services: -**Vector Space Homework | Vector Space Homework Help | Vector Space Homework Help Services | Live Vector Space Homework Help | Vector Space Homework Tutors | Online Vector Space Homework Help | Vector Space Tutors | Online Vector Space Tutors | Vector Space Homework Services | Vector SpaceSubmit Your Query ???

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