Vector space ( Definition ) : Linear Algebra

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Linear Algebra

Vector Spaces

Vector Space ( Definition ) :

Let V be a set of vectors and F be a field of scalars. Then V is said to be a vector space

under the field of scalars F if

I) (V,+) is abelian

II) The scalar multiplication condition exists in V

i.e. for a, 𝞊 F and 𝜶 𝞊 V ⇒ a𝜶 𝞊 V

III) together with the following 4 inner conditions exists in V.

i) a ( 𝜶 + 𝛃 ) = a𝜶 + a𝛃 for a 𝞊 F and 𝜶, 𝛃 𝞊 F.

ii) (a+b)𝜶 = a𝜶 +b𝜶 for a,b 𝞊 F and 𝜶 𝞊 V

iii) (ab)𝜶 = a(b𝜶) for a,b 𝞊 F and 𝜶 𝞊 V

iv) 1.𝜶 = 𝜶 for 𝜶 𝞊 V

Note :

* Here the scalar multiplication condition is called external composition

* Here the vector addition is called an internal composition.

Examples :

1. The Real line ℛ

2. The Euclidean plane ℛ ²

3. The Euclidian spaces ℛ ³, ℛ ⁴…, ℛ ⁿ,…

4. The set of 2x2 matrices etc .

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