devotion & mathematics

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 : * 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 ⇒  a𝜶 +b𝞫 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 :                                        * 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 ...

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:            * 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 an...

For a video explanation , click the link 👉 https://youtu.be/jfKdIWXVYsQ 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 is  𝜶 𝞊 W, 𝞫 𝞊 W ⇒  a𝜶 +b𝞫 𝞊 W proof :        * 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/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...