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 ⇒  ...

  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, 𝞫 𝞊 W ⇒  𝜶 - 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...

Linear Combination of vectors :                      Suppose  𝜶 1 , 𝜶 2 , … , 𝜶 n   be any n  vectors in a vector space V ( F ) . Then for  some scalars   the representation  a 1 , a 2 , … , a n   the representation    a 1 𝜶 1 +a 2 𝜶 2 +   … +a n   𝜶 n  is  called a linear  combination of vectors    𝜶 1 , 𝜶 2 , … , 𝜶 n . * 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 here 👉 https://youtu.be/DTwC9rO2rWQ Theorem :               If  W 1 and   W 2 be two subspaces of the vector space V(F) . Then         1 )    W 1 + W 2   is a subspace of V(F)      and        2 )   W 1  ⊆    W 1 + W 2    and       W 2    ⊆  W 1 + W 2      Proof :                                        * Linear combination of vectors    *   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 b...

Linear Sum of Subspaces: Definition                                 Let   W 1 and   W 2 be two subspaces of the vector space V(F) . Then the linear sum  of the subspaces W 1 & W 2 , denoted by W 1 + W 2 , is the set of all sums 𝜶 1 + 𝜶 2 such  that   𝜶 1 𝞊 W 1 ,  𝜶 2 𝞊 W 2 i.e.  W 1 + W 2 = { 𝜶 1 + 𝜶 2 / 𝜶 1 𝞊 W 1 ,  𝜶 2 𝞊 W 2 }.   * LInear combination of vectors  * 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                                 ...