devotion & mathematics

 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                                                   ...

  OPEN & CLOSED SETS In this session we see open & closed sets with their properties. OPEN SET : -      A non empty set A in the real line R is called an open set , every point in A is an interior point of A.                                                                     Examples:- * (a, b)  is an open set.  * [a, b] is not open since the end points a, b are not interior points. * Every finite set is not open. * The sets N, Z and Q are not open sets. * The set R of all real numbers is open. * The set of complex numbers |z|<1 is open. * The set of complex numbers |z|  ≤1 is not open. * Every neighbourhood is an open set. Properties of open sets:- 1. Every open set is a union of open intervals. 2. Union of two open sets is open. 3. Intersection of two open sets is ...