devotion & mathematics

      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  * 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 ...

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 : *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 ⇒...

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 ...

Exercise 1(c) : Problems & Solutions                   1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:    (i) { 2, 3, 4 } ... { 1, 2, 3, 4, 5 }       Solution :         Given sets { 2, 3, 4 } and  { 1, 2, 3, 4, 5 }         Since the set { 1, 2, 3, 4, 5 } containing all the elements  2, 3, 4 , we have         { 2, 3, 4 } ⊂ { 1, 2, 3, 4, 5 }    (ii) { a, b, c } ... { b, c, d }        Solution :          Given sets are { a, b, c } & { b, c, d }.           Since ' a ' is not in the set { b, c, d }, we have              { a, b, c } ⊄ { b, c, d }    (iii) {x : x is a student of Class XI of your school} ... {x : x student of your school}        Solutiion :   ...

For a video explanation, click the link 👉 https://youtu.be/Rtwx1-VKBQo Theorem :                    A non-empty set W is a subset  of a vector space V(F)  . W is a subspace of V                  if and only if a 𝞊 F and 𝝰 , 𝞫 𝞊 W ⇒ a𝝰 + 𝞫 𝞊 W .  Proof:            *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 ...