devotion & mathematics

For a video explanation, click here 👉 https://youtu.be/b0KvzStFHAc   Linearly Independent Vectors :          Let 𝜶 1 , 𝜶 2 , … , 𝜶 n be any n vectors in a vector space V ( F ). Then the      vectors 𝜶 1 , 𝜶 2 ,  … , 𝜶 n are   called Linearly Independent vectors , if there exists       Scalars a 1 , a 2 , … , a n   such that  a 1 𝜶 1 + a 2 𝜶 2 + … + a n 𝜶 n = 0 implies a 1 = a 2 = … = a n = 0.      Linearly Dependent Vectors :          Let 𝜶 1 , 𝜶 2 , … , 𝜶 n be any n vectors in a vector space V ( F ). Then the      vectors 𝜶 1 , 𝜶 2 , … , 𝜶 n are   called Linearly Dependent vectors , if there exists scalars      a 1 , a 2 , … , a n  not all zero  such that   a 1 𝜶 1 + a 2 𝜶 2 + … + a n 𝜶 n = 0   

       Theorem :      A singleton set with non-zero vector is Linearly Independent   Proof :           Suppose V ( F) is a vector space and S = {𝜶} is a subset of V where 𝜶 ≠ 0 .         Now we prove S is Linearly Independent.          For a 𝞊 F such that a𝜶 = 0                                      ⇒ a=0 or 𝜶 = 0  .          Since 𝜶 ≠ 0 , we have a = 0.            ∴ S is Linearly Independent.                                               ***Hence the proof ***      Explanation :           Linearly independent set means if the linear combination of vectors is ...

For a video explanation , click here 👉   https://youtu.be/8T7JZOyRO70   Theorem :       Let V ( F ) be a vector space and S = { 𝜶 1 , 𝜶 2 , … , 𝜶 n } is a finite subset of non – zero vectors of V ( F ). Then S is linearly dependent if and only if some vector 𝜶 k 𝟄 S , 2 ≤ k ≤ n , can be expressed as a linear combination of its preceeding vectors. Proof :   * L ( S ) = L ( S' ) * linear sum of subspaces * What is a vector space * linear span of a set    * 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                                                  ...