devotion & mathematics

Theorem :      If W 1 and W 2 are any two subspace of a vector space V ( F ) then            L ( W 1 ⋃ W 2  ) = W 1 + W 2 . Proof :          * Linear combination of vectors * linear sum of subspaces * What is a vector space  * Theorem on vector space      * 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. * Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W      to be a subspace of V is  𝜶 𝞊 W, 𝞫 𝞊 W ⇒  ...

 Linear Span of a set :                                                   Let S be a non-empty subset of a vector space V (F). Then the set of  all  linear combinations of vectors of S is called the linear span of S and is denoted by L (S).          i.e. L (S) = { 𝜶 / 𝜶 = 𝜮 , 1 ≤ i ≤ n } . For example :             If S = {  𝜶 1 , 𝜶 2 , … , 𝜶 n .} and if 𝜶 𝞊 L (S) then there exist scalars  a 1 , a 2 , … , a n   such            that    𝜶 =  a 1 𝜶 1 +a 2 𝜶 2 +   … +a n 𝜶 n . * Linear combination of vectors * linear sum of subspaces * What is a vector space  * Theorem on vector space      * Historical Introduction to Linear Algebra   * Let V(F) be a vector space and let W ⊆ V. The necess...