devotion & mathematics

  For a video explanation , click the link 👉 https://youtu.be/xxeVZKw8MAc Theorem :             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.  Proof :             * External Composition   * 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 is  𝜶 𝞊 W, 𝞫 𝞊 W ⇒  a𝜶 +b𝞫 𝞊 W # Let V(F) be a vector space and let W...

For a video explanation , click the link 👉 https://youtu.be/ukbd5oWXMH4                        Vector Subspace : Definition         Let V(F) be a vector space and W ⊆ V. Then W is said to be a subspace of V if W             is    itself a  vector space over F with the same operation of vector addition and                  scalar   multiplication in V.  Example :          * The set of 2x2 triangular matrices is a vector subsapce of the space of all 2x2            with   matrices  with real entries.  * External Composition     * 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...

   Exercise 1(b) : ( problems & solutions ) :                                      1. Which of the following are examples of the null set .      (i) Set of odd natural numbers divisible by 2.        Solution :            The odd natural numbers are 1,3,5,...             Let A be the set of odd natural numbers divisible by 2.             We know no any odd number is divisible by 2.            ∴  A has no elements.             Hence A is null set.      (ii) Set of even prime numbers .         Solution :                The prime numbers are 2,3,5,7,11,13,...               Let A be the ...

For a video explanation , click the link   https://youtu.be/YB_p6iPSIfQ    Theorem :             Let V(F) be a vector space and 0 and  O be the zero scalar and zero vector respectively. Then    (i)  a O = O ∀ a 𝞊 F.   (ii) o 𝜶 = O ∀ a 𝞊 F, 𝜶 𝞊 V     (iii) a ( - 𝜶 ) = - (a𝜶) = (-a) 𝜶 ∀ a 𝞊 F , 𝜶 𝞊 V   (iv) a ( 𝜶 - 𝛃 ) = a𝜶 - a𝛃 ∀ a 𝞊 F , 𝜶,𝛃 𝞊 V.    Proof :                            * External Composition   *  Historical Introduction to Linear Algebra  * What is a vector space *   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. #vector #algebra #vector algebra   #iit #jee #mains #dimens...

                                                              A  historical introduction to Linear Algebra explains how ideas about  solving equations, geometry, and arrays of numbers gradually developed into the modern subject we  now call linear algebra. Historical Introduction to Linear Algebra 1. Early Origins – Ancient Civilizations The roots of linear algebra go back thousands of years. Ancient civilizations such as Egypt and China used methods equivalent to solving systems of linear equations. Around 200 BCE , the Chinese mathematical text The Nine Chapters on the Mathematical Art described procedures for solving simultaneous equations using tables of numbers. These methods resemble what is now called Gaussian elimination . 2. Development of Analytic Geometry (17th Century) A major step occurred during the 1600s...

 External Composition :          An external composition means it is a binary operation performed between two different kinds of elements.   For example :      In the definition of vector space , The scalar multiplication condition is an external composition because  the scalar multiplication condition is                   for a, 𝞊 F and 𝜶 𝞊  V ⇒   a𝜶 𝞊  V     In this condition the product  a𝜶 performed between a scalar  a  and a vector  𝜶                                          *** * The fourth dimensional objects * what is a dimension of an object?   * The Euclidean space   ℛ n * Historical Introduction to Linear Algebra   * Theorem on vector space    *   Let V(F) be a vector spac...

 [ To see video explanation click on  👉 https://youtu.be/Ojr-skYtVi4 ]                                                                                      Linear Algebra                                     Vector Spaces  Vector Space ( Definition ) :               Let V be a set of vectors and F be a field of scalars. Then V is said to be a vector space  under the field of scalars F if  I)  (V,+) is abelian II) The scalar multiplication condition exists in V       i.e. for  a, 𝞊 F and 𝜶 𝞊  V ⇒   a𝜶 𝞊  V III) together with the following 4 inner conditions exists in V. ...