devotion & mathematics

For a video explanation, click here 👉 https://youtu.be/r3u7AVlNg-Y Theorem : Let S = { 𝜶1 , 𝜶2 , ... , 𝜶n } be a subset of the vector space V ( F ) . If 𝜶i in S is a linear combination of its preceeding vectors then L ( S ) = L ( S' ) where S' = { 𝜶1 , 𝜶2 , ... , 𝜶(i-1) , 𝜶(i+1) , ... , 𝜶n } . Proof :  * Theorem on Linear Dependence  * 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                                                      (ii) a 𝞊 F , 𝜶 𝞊 W ⇒ a𝜶 𝞊 W. * Let V(F) be a vector ...

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                                                  ...

For a video explanation, click the link 👉 https://youtu.be/NuvBv7ppF2k Theorem :         For any subset S of a vector space V ( F ) , the linear span of S L ( S ) is a subspace of V.   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 suf...

For a video explanation , click here 👉 https://youtu.be/DvN0WBjcOjY 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  ...

For a video explanation, click here 👉 https://youtu.be/BdD_ET3MfoA   Problems on linear combination :  Problem 1 :                   Express the vector 𝜶 = (1 , -2 , 5 ) as a linear combination of the vectors e 1 = ( 1 , 1 , 1 ) , e 2 = ( 1 , 2 , 3 ) , e 3 = ( 2 , -1 , 1 ). Solution : Problem 2 : Show that the vector 𝜶 = ( 2 , -5 , 3 ) in R 3 can not be expressed as a linear combination of the vectors e 1 = ( 1 , -3 , 2 ) ; e 2 = ( 2 , -4 , -1 ) ; e 3 = ( 1 , -5 , 7 ) Solution : * 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,...

 For a video explanation, click here 👉 https://youtu.be/T5oscsd_bTc  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 L...