Vector space ( Definition ) : Linear Algebra : Degree

 [ To see video explanation click on  👉https://youtu.be/Ojr-skYtVi4 ]

                                  

                                                 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 .

                                                               ***

* External Composition 

 * Theorem on vector space  

* The fourth dimensional objects

* what is a dimension of an object? 

* The Euclidean space  ℛ ⁿ

* Historical Introduction to Linear Algebra

*  Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W 

 to be a subspace of V are (i) 𝜶 𝞊 W, 𝞫 𝞊 W ⇒  𝜶 - 𝞫 𝞊 W   (ii) a 𝞊 F , 𝜶 𝞊 W ⇒ a𝜶 𝞊 W.

                                                  

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