Publish On: 2019-03-17

# jollay

Total Post: 559

## Question: Explain vector space in algebra

Reply On: 2013-10-08

# Mjay Jollay

Total Post: 0

## ANS: Explain vector space in algebra

Let < 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

V

V

a + 0 = a = 0 + a ∀ a V

then, 0 is called the identity element.

V

a + b = b + a = 0.

then b is known as the inverse of a and vice versa.

V

(a) = ()a ∀ , F and V.

V

V

(a + b) = a + b, ∀ a, b V and ∀ F.

V

( +)a = a + a ∀ , F and ∀ a V

V

1.a = a = a.1; ∀ a V.

The elements of F are known as scalars and the elements of V are called vectors.

V

_{1}: Closure property: ∀ a, b V a + b V.V

_{2}: Associative law: ∀ a, b, c V (a + b) + c = a (b + c).V

_{3}: Existence of identity: There exists an element 0 V such thata + 0 = a = 0 + a ∀ a V

then, 0 is called the identity element.

V

_{4}: Existence of inverse: ∀ a V, there exists b V such thata + b = b + a = 0.

then b is known as the inverse of a and vice versa.

V

_{5}: Scalar multiplication is associative, i.e.(a) = ()a ∀ , F and V.

V

_{6}: Commutative law: a + b = b + aV

_{7}: Scalar multiplication is distributive over addition in F.(a + b) = a + b, ∀ a, b V and ∀ F.

V

_{8}: Distributivity of scalar multiplication over addition in F.( +)a = a + a ∀ , F and ∀ a V

V

_{9}: Property of unity Let 1 F be the unity of F, then1.a = a = a.1; ∀ a V.

The elements of F are known as scalars and the elements of V are called vectors.

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