devotion & mathematics

For a video explanation, click here 👉 https://youtu.be/j9Zc28mgKwk 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                                 ...

 For a video explanation, click the link 👉 https://youtu.be/qSTyvyJliw4     Theorem :                  The union of subspaces of a vector space is again a subspace if and only if one is contained in  the other.   Proof :                                   * Linear Sum of Subspaces   * What is a vector space * Linear combination of vectors   * 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/Ai7fvMYv4Jo Theorem :            The intersection of subspaces of  a vector space is again a subspace of the vector                space                                           Or          If    W 1  and  W 2  be any two subspaces of a vector spave V(F) then    W 1 ∩ W 2    is also             a subspace of V(F)  Proof :                                                     * Linear Sum of Subspaces   * What is a vector space  * Theorem on vector space      * Historical Introducti...

For a video explanation ,click on 👉 https://youtu.be/JHbCJAh61eM Problem 2 : Let p,q,r be the fixed elements of a field F. Show that the set W of all triads (x,y,z) of elements of F, such that px+qy+rz = 0 is a vector subspace of V3( F ) Solution :  * Problems on linear combination * 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...

For a video explanation, click the link 👉   https://youtu.be/FdPspiSjc-c            Problem :          The set W of ordered triads ( x , y , 0 ) where x , y 𝞊 F  is a subspace of     V 3 (F). Solution :                   *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 ...